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{{Other uses|Matrix (disambiguation){{!}}Matrix}}
{{Short description|Two-dimensional array of numbers with specific operations}}
{{Redirect|Matrix theory|the physics topic|Matrix string theory}}
[[File:Matris.png|thumb|An {{math|''m'' &times; ''n''}} matrix: the {{math|''m''}} rows are horizontal and the {{math|''n''}} columns are vertical. Each element of a matrix is often denoted by a variable with two [[index notation|subscripts]]. For example, {{math|''a''<sub>2,1</sub>}} represents the element at the second row and first column of the matrix.]]

In [[mathematics]], a '''matrix''' (plural: '''matrices''') is a [[rectangle|rectangular]] ''[[wikt:array|array]]''<ref>Equivalently, ''[[wikt:table|table]]''.</ref> (see [[irregular matrix]]) of [[number]]s, [[symbol (formal)|symbol]]s, or [[expression (mathematics)|expression]]s, arranged in ''[[wikt:row|row]]s'' and ''[[wikt:column|column]]s''.<ref>{{harvtxt|Anton|1987|p=23}}</ref><ref>{{harvtxt|Beauregard|Fraleigh|1973|p=56}}</ref> For example, the dimension of the matrix below is 2 × 3 (read "two by three"), because there are two rows and three columns:
:<math>\begin{bmatrix}1 & 9 & -13 \\20 & 5 & -6 \end{bmatrix}.</math>
Provided that they have the same size (each matrix has the same number of rows and the same number of columns as the other), two matrices can be [[matrix addition|added]] or subtracted element by element (see [[Conformable matrix]]). The rule for [[matrix multiplication]], however, is that ''two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second'' (i.e., the inner dimensions are the same, '''''n''''' for an (''m''×'''''n''''')-matrix times an ('''''n'''''×''p'')-matrix, resulting in an (''m''×''p'')-matrix. There is no product the other way round, a first hint that matrix multiplication is not [[commutative property|commutative]]. Any matrix can be [[matrix multiplication#Product with a scalar|multiplied]] element-wise by a [[scalar (mathematics)|scalar]] from its associated [[field (mathematics)|field]].

The individual items in an ''m''×''n'' matrix '''A''', often denoted by ''a''<sub>''i'',''j''</sub>, where ''i'' and ''j'' usually vary from {{math|1}} to ''m'' and ''n'', respectively, are called its ''elements'' or ''entries''.<ref>{{cite book|last1=Young|first1=Cynthia|title=Precalculus|publisher=Laurie Rosatone|page=727}}</ref> For conveniently expressing an element of the results of matrix operations the indices of the element are often attached to the parenthesized or bracketed matrix expression; e.g.: ('''AB''')<sub>i,j</sub> refers to an element of a matrix product. In the context of [[abstract index notation]] this ambiguously refers also to the whole matrix product.

A major application of matrices is to represent [[linear transformation]]s, that is, generalizations of [[linear function]]s such as {{nowrap|''f''(''x'') {{=}} 4''x''}}. For example, the [[rotation (mathematics)|rotation]] of [[Euclidean vector|vector]]s in three-[[dimension]]al space is a linear transformation, which can be represented by a [[rotation matrix]] '''R''': if '''v''' is a [[column vector]] (a matrix with only one column) describing the [[position (vector)|position]] of a point in space, the product '''Rv''' is a column vector describing the position of that point after a rotation. The product of two [[transformation matrices]] is a matrix that represents the [[function composition|composition]] of two [[transformation (function)|transformations]]. Another application of matrices is in the solution of [[system of linear equations|systems of linear equations]]. If the matrix is [[#Square matrices|square]], it is possible to deduce some of its properties by computing its [[determinant]]. For example, a square matrix [[invertible matrix|has an inverse]] [[if and only if]] its determinant is not [[zero]]. Insight into the [[geometry]] of a linear transformation is obtainable (along with other information) from the matrix's [[eigenvalues and eigenvectors]].

Applications of matrices are found in most scientific fields. In every branch of [[physics]], including [[classical mechanics]], [[optics]], [[electromagnetism]], [[quantum mechanics]], and [[quantum electrodynamics]], they are used to study physical phenomena, such as [[rigid-body dynamics|the motion of rigid bodies]]. In [[computer graphics]], they are used to manipulate [[3D models]] and project them onto a [[glossary of computer graphics#screen space|2-dimensional screen]]. In [[probability theory]] and [[statistics]], [[stochastic matrix|stochastic matrices]] are used to describe sets of probabilities; for instance, they are used within the [[PageRank]] algorithm that ranks the pages in a Google search.<ref>K. Bryan and T. Leise. The $25,000,000,000 eigenvector: The linear algebra behind Google. SIAM Review, 48(3):569–581, 2006.</ref> [[Matrix calculus]] generalizes classical [[mathematical analysis|analytical]] notions such as [[derivative]]s and [[Exponential function|exponentials]] to higher dimensions. Matrices are used in [[economics]] to describe systems of economic relationships.

A major branch of [[numerical analysis]] is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. [[#Decomposition|Matrix decomposition methods]] simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as [[sparse matrix|sparse matrices]] and [[diagonal matrix|near-diagonal matrices]], expedite computations in [[finite element method]] and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the [[derivative (calculus)|derivative]] operator, which acts on the [[Taylor series]] of a function.

==Definition==
A ''matrix'' is a rectangular array of [[number]]s or other mathematical objects for which operations such as [[#Basic operations|addition]] and [[#Matrix multiplication|multiplication]] are defined.<ref>{{Harvard citations |author=Lang |year=2002 |nb=yes}}</ref> Most commonly, a matrix over a [[field (mathematics)|field]] ''F'' is a rectangular array of scalars each of which is a member of ''F''.<ref>{{harvtxt|Fraleigh|1976|p=209}}</ref><ref>{{harvtxt|Nering|1970|p=37}}</ref> Most of this article focuses on ''real'' and ''complex matrices'', that is, matrices whose elements are [[real number]]s or [[complex number]]s, respectively. More general types of entries are discussed [[#Matrices with more general entries|below]]. For instance, this is a real matrix:
:<math>\mathbf{A} = \begin{bmatrix}
-1.3 &  0.6 \\
20.4 &  5.5 \\
 9.7 & -6.2
\end{bmatrix}.</math>

The numbers, symbols, or expressions in the matrix are called its ''entries'' or its ''elements''. The horizontal and vertical lines of entries in a matrix are called ''rows'' and ''columns'', respectively.

===Size===
The size of a matrix is defined by the number of rows and columns that it contains. A matrix with ''m'' rows and ''n'' columns is called an ''m''&nbsp;×&nbsp;''n'' matrix or ''m''-by-''n'' matrix, while ''m'' and ''n'' are called its ''dimensions''. For example, the matrix '''A''' above is a 3&nbsp;×&nbsp;2 matrix.

Matrices with a single row are called ''[[row vector]]s'', and those with a single column are called ''[[column vectors]]''. A matrix with the same number of rows and columns is called a ''[[square matrix]]''. A matrix with an infinite number of rows or columns (or both) is called an [[#Infinite matrices|''infinite matrix'']]. In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an [[#Empty matrices|''empty matrix'']].

{| class="wikitable"
|-
! Name !! Size !! Example !! Description
|-
| [[Row vector]] || 1&nbsp;×&nbsp;''n'' || style="text-align:center;" | <math>\begin{bmatrix}3 & 7 & 2 \end{bmatrix}</math>
| A matrix with one row, sometimes used to represent a vector
|-
| [[Column vector]] || ''n''&nbsp;×&nbsp;1 || style="text-align:center;" | <math>\begin{bmatrix}4 \\ 1 \\ 8 \end{bmatrix}</math>
| A matrix with one column, sometimes used to represent a vector
|-
| [[Square matrix]] || ''n''&nbsp;×&nbsp;''n'' || style="text-align:center;" | <math>\begin{bmatrix}
9 & 13 & 5 \\
1 & 11 & 7 \\
2 &  6 & 3
\end{bmatrix}</math>
| A matrix with the same number of rows and columns, sometimes used to represent a [[#Linear transformations|linear transformation]] from a vector space to itself, such as [[reflection (mathematics)|reflection]], [[rotation (mathematics)|rotation]], or [[shear mapping|shearing]].
|}

==Notation==
Matrices are commonly written in [[box bracket]]s or [[parentheses]]:
:<math>\mathbf{A} =
\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix} =
\begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{pmatrix}=\left(a_{ij}\right) \in \mathbb{R}^{m \times n}.
</math>

The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are usually symbolized using [[upper-case]] letters (such as '''A''' in the examples above), while the corresponding [[lower-case]] letters, with two subscript indices (for example, ''a''<sub>11</sub>, or ''a''<sub>1,1</sub>), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special [[emphasis (typography)|typographical style]], commonly boldface upright (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, (for example, <math>\underline{\underline{A}}</math>).

The entry in the ''i''-th row and ''j''-th column of a matrix '''A''' is sometimes referred to as the ''i'',''j'', (''i'',''j''), or (''i'',''j'')th entry of the matrix, and most commonly denoted as ''a''<sub>''i'',''j''</sub>, or ''a<sub>ij</sub>''. Alternative notations for that entry are ''A''[''i,j''] or ''A''<sub>''i,j''</sub>. For example, the (1,3) entry of the following matrix '''A''' is 5 (also denoted ''a''<sub>13</sub>, ''a''<sub>1,3</sub>, ''A''[''1,3''] or ''A''<sub>''1,3''</sub>):
:<math>\mathbf{A}=\begin{bmatrix}
 4 & -7 & \color{red}{5} & 0 \\
-2 &  0 & 11 & 8 \\
19 &  1 & -3 & 12
\end{bmatrix}</math>

Sometimes, the entries of a matrix can be defined by a formula such as ''a''<sub>''i'',''j''</sub> = ''f''(''i'', ''j''). For example, each of the entries of the following matrix '''A''' is determined by ''a<sub>ij</sub>'' = ''i'' − ''j''.
:<math>\mathbf A = \begin{bmatrix}
0 & -1 & -2 & -3\\
1 &  0 & -1 & -2\\
2 &  1 &  0 & -1
\end{bmatrix}</math>
In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parentheses. For example, the matrix above is defined as '''A''' = [''i''−''j''], or '''A''' = ((''i''−''j'')). If matrix size is ''m'' × ''n'', the above-mentioned formula ''f''(''i'', ''j'') is valid for any ''i'' = 1, ..., ''m'' and any ''j'' = 1, ..., ''n''. This can be either specified separately, or using ''m'' × ''n'' as a subscript. For instance, the matrix '''A''' above is 3 × 4 and can be defined as '''A''' = [''i'' − ''j''] (''i'' = 1, 2, 3; ''j'' = 1, ..., 4), or '''A''' = [''i'' − ''j'']<sub>''3''×''4''</sub>.

Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an ''m''-×-''n'' matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an ''m''-by-''n'' matrix are indexed by {{nowrap|0 ≤ ''i'' ≤ ''m'' − 1}} and {{nowrap|0 ≤ ''j'' ≤ ''n'' − 1}}.<ref>{{Harvard citations |last1=Oualline |year=2003 |loc=Ch. 5 |nb=yes}}</ref> This article follows the more common convention in mathematical writing where enumeration starts from 1.

An asterisk is occasionally used to refer to whole rows or columns in a matrix. For example, ''a''<sub>''i'',∗</sub> refers to the i<sup>th</sup> row of '''A''', and ''a''<sub>∗,''j''</sub> refers to the j<sup>th</sup> column of '''A'''. The [[set (mathematics)|set]] of all ''m''-by-''n'' matrices is denoted 𝕄(''m'', ''n'').

==Basic operations==
{{External media |width=210px |align=right |video1=[https://ed.ted.com/lessons/how-to-organize-add-and-multiply-matrices-bill-shillito How to organize, add and multiply matrices - Bill Shillito], [[TED (conference)|TED ED]]<ref name="Ted">{{cite web |title=How to organize, add and multiply matrices - Bill Shillito |work= |publisher=[[TED (conference)|TED ED]] |date= |url=https://ed.ted.com/lessons/how-to-organize-add-and-multiply-matrices-bill-shillito |accessdate=April 6, 2013}}</ref>}}
There are a number of basic operations that can be applied to modify matrices, called ''matrix addition'', ''scalar multiplication'', ''transposition'', ''matrix multiplication'', ''row operations'', and ''submatrix''.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)}}</ref>

===Addition, scalar multiplication, and transposition===
{{Main|Matrix addition|Scalar multiplication|Transpose}}

{| class="wikitable"
|-
! Operation
! Definition
! Example
|-
| [[Matrix addition|Addition]]
| The ''sum'' '''A'''+'''B''' of two ''m''-by-''n'' matrices '''A''' and '''B''' is calculated entrywise:
:('''A''' + '''B''')<sub>''i'',''j''</sub> = '''A'''<sub>''i'',''j''</sub> + '''B'''<sub>''i'',''j''</sub>, where 1 ≤ ''i'' ≤ ''m'' and 1 ≤ ''j'' ≤ ''n''.
|
<math>

\begin{bmatrix}
1 & 3 & 1 \\
1 & 0 & 0
\end{bmatrix}
+
\begin{bmatrix}
0 & 0 & 5 \\
7 & 5 & 0
\end{bmatrix}
=
\begin{bmatrix}
1+0 & 3+0 & 1+5 \\
1+7 & 0+5 & 0+0
\end{bmatrix}
=
\begin{bmatrix}
1 & 3 & 6 \\
8 & 5 & 0
\end{bmatrix}
</math>
|-
| [[Scalar multiplication]]
| The product ''c'''''A''' of a number ''c'' (also called a [[scalar (mathematics)|scalar]] in the parlance of [[abstract algebra]]) and a matrix '''A''' is computed by multiplying every entry of '''A''' by ''c'':
:{{nowrap begin}}(''c'''''A''')<sub>''i'',''j''</sub> = ''c'' · '''A'''<sub>''i'',''j''</sub>.{{nowrap end}}
This operation is called ''scalar multiplication'', but its result is not named "scalar product" to avoid confusion, since "scalar product" is sometimes used as a synonym for "[[inner product]]".
| <math>2 \cdot

\begin{bmatrix}
1 &  8 & -3 \\
4 & -2 &  5
\end{bmatrix}
=
\begin{bmatrix}
2 \cdot 1 & 2\cdot 8 & 2\cdot -3 \\
2\cdot 4 & 2\cdot -2 & 2\cdot 5
\end{bmatrix}
=
\begin{bmatrix}
2 & 16 & -6 \\
8 & -4 & 10
\end{bmatrix}
</math>
|-
| [[Transpose|Transposition]]
| The ''transpose'' of an ''m''-by-''n'' matrix '''A''' is the ''n''-by-''m'' matrix '''A'''<sup>T</sup> (also denoted '''A'''<sup>tr</sup> or <sup>t</sup>'''A''') formed by turning rows into columns and vice versa:
:{{nowrap begin}}('''A'''<sup>T</sup>)<sub>''i'',''j''</sub> = '''A'''<sub>''j'',''i''</sub>.{{nowrap end}}
| <math>

\begin{bmatrix}
1 &  2 & 3 \\
0 & -6 & 7
\end{bmatrix}^\mathrm{T} =
\begin{bmatrix}
1 &  0 \\
2 & -6 \\
3 &  7
\end{bmatrix}
</math>
|}

Familiar properties of numbers extend to these operations of matrices: for example, addition is [[commutative]], that is, the matrix sum does not depend on the order of the summands: '''A'''&nbsp;+&nbsp;'''B'''&nbsp;=&nbsp;'''B'''&nbsp;+&nbsp;'''A'''.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem I.2.6}}
</ref>
The transpose is compatible with addition and scalar multiplication, as expressed by (''c'''''A''')<sup>T</sup> = ''c''('''A'''<sup>T</sup>) and ('''A'''&nbsp;+&nbsp;'''B''')<sup>T</sup>&nbsp;=&nbsp;'''A'''<sup>T</sup>&nbsp;+&nbsp;'''B'''<sup>T</sup>. Finally, ('''A'''<sup>T</sup>)<sup>T</sup>&nbsp;=&nbsp;'''A'''.
<!--Another, much less often used notion of matrix addition is the [[direct sum (matrix)|direct sum]].-->

===Matrix multiplication===
{{Main|Matrix multiplication}}
[[File:MatrixMultiplication.png|thumb|300px|Schematic depiction of the matrix product '''AB''' of two matrices '''A''' and '''B'''.]]
''Multiplication'' of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If '''A''' is an ''m''-by-''n'' matrix and '''B''' is an ''n''-by-''p'' matrix, then their ''matrix product'' '''AB''' is the ''m''-by-''p'' matrix whose entries are given by [[dot product]] of the corresponding row of '''A''' and the corresponding column of '''B''':
:<span id="matrix_product"><math>[\mathbf{AB}]_{i,j} = a_{i,1}b_{1,j} + a_{i,2}b_{2,j} + \cdots + a_{i,n}b_{n,j} = \sum_{r=1}^n a_{i,r}b_{r,j},</math></span>

where 1 ≤ ''i'' ≤ ''m'' and 1 ≤ ''j'' ≤ ''p''.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.2.20}}</ref> For example, the underlined entry 2340 in the product is calculated as {{math|(2 × 1000) + (3 × 100) + (4 × 10) {{=}} 2340:}}
:<math>
\begin{align}
\begin{bmatrix}
\underline{2} & \underline 3 & \underline 4 \\
1 & 0 & 0 \\
\end{bmatrix}

\begin{bmatrix}
0 & \underline{1000} \\
1 & \underline{100} \\
0 & \underline{10} \\
\end{bmatrix}
&=
\begin{bmatrix}
3 & \underline{2340} \\
0 & 1000 \\
\end{bmatrix}.
\end{align}
</math>

Matrix multiplication satisfies the rules ('''AB''')'''C''' = '''A'''('''BC''') ([[associativity]]), and ('''A''' + '''B''')'''C''' = '''AC''' + '''BC''' as well as '''C'''('''A''' + '''B''') = '''CA''' + '''CB''' (left and right [[distributivity]]), whenever the size of the matrices is such that the various products are defined.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem I.2.24}}</ref> The product '''AB''' may be defined without '''BA''' being defined, namely if '''A''' and '''B''' are ''m''-by-''n'' and ''n''-by-''k'' matrices, respectively, and {{nowrap|''m'' ≠ ''k''.}} Even if both products are defined, they need not be equal, that is, generally
:'''AB''' ≠ '''BA''',

that is, <span id="non_commutative">matrix multiplication is not [[commutative property|commutative]],</span> in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:
:<math>\begin{bmatrix}
1 & 2\\
3 & 4\\
\end{bmatrix}

\begin{bmatrix}
0 & 1\\
0 & 0\\
\end{bmatrix}=
\begin{bmatrix}
0 & 1\\
0 & 3\\
\end{bmatrix},
</math>
whereas
:<math>\begin{bmatrix}
0 & 1\\
0 & 0\\
\end{bmatrix}

\begin{bmatrix}
1 & 2\\
3 & 4\\
\end{bmatrix}=
\begin{bmatrix}
3 & 4\\
0 & 0\\
\end{bmatrix}.
</math>

Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the [[Hadamard product (matrices)|Hadamard product]] and the [[Kronecker product]].<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |loc=Ch. 4 and 5 |nb=yes}}</ref> They arise in solving matrix equations such as the [[Sylvester equation]].

===Row operations===
{{Main|Row operations}}
There are three types of row operations:
# row addition, that is adding a row to another.
# row multiplication, that is multiplying all entries of a row by a non-zero constant;
# row switching, that is interchanging two rows of a matrix;
These operations are used in a number of ways, including solving [[linear equation]]s and finding [[matrix inverse]]s.

==={{anchor|Submatrix}} Submatrix===
A '''submatrix''' of a matrix is obtained by deleting any collection of rows and/or columns.<ref>{{harvtxt|Bronson|1970|p=16}}</ref><ref>{{harvtxt|Kreyszig|1972|p=220}}</ref><ref name="Protter 1970 869">{{harvtxt|Protter|Morrey|1970|p=869}}</ref> For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2:
:<math>\mathbf{A}=\begin{bmatrix}
1 & \color{red}{2} & 3 & 4 \\
5 & \color{red}{6} & 7 & 8 \\
\color{red}{9} & \color{red}{10} & \color{red}{11} & \color{red}{12}
\end{bmatrix} \rightarrow \begin{bmatrix}
1 & 3 & 4 \\
5 & 7 & 8
\end{bmatrix}.
</math>

The [[minor (linear algebra)|minors]] and cofactors of a matrix are found by computing the [[determinant]] of certain submatrices.<ref name="Protter 1970 869"/><ref>{{harvtxt|Kreyszig|1972|pp=241,244}}</ref>

A '''principal submatrix''' is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain.<ref>{{Citation |title=Matrices and Linear Algebra|series=Dover Books on Mathematics|first1=Hans|last1=Schneider|first2=George Phillip|last2=Barker|publisher=Courier Dover Corporation|year=2012|isbn=9780486139302|page=251|url=https://books.google.com/books?id=9vjBAgAAQBAJ&pg=PA251}}.</ref><ref>{{Citation |title=Theory of Matrices|series=Dover books on advanced mathematics|first=Sam|last=Perlis|publisher=Courier Dover Corporation|year=1991|isbn=9780486668109|page=103|url=https://books.google.com/books?id=5_sxtcnvLhoC&pg=PA103}}.</ref> Other authors define a principal submatrix as one in which the first ''k'' rows and columns, for some number ''k'', are the ones that remain;<ref>{{Citation |title=Elementary Linear Algebra|first=Howard|last=Anton|edition=10th|publisher=John Wiley & Sons|year=2010|isbn=9780470458211|page=414|url=https://books.google.com/books?id=YmcQJoFyZ5gC&pg=PA414}}.</ref> this type of submatrix has also been called a '''leading principal submatrix'''.<ref>{{Citation |title=Matrix Analysis|first1=Roger A.|last1=Horn|first2=Charles R.|last2=Johnson|edition=2nd|publisher=Cambridge University Press|year=2012|isbn=9780521839402|page=17|url=https://books.google.com/books?id=5I5AYeeh0JUC&pg=PA17}}.</ref>

==Linear equations==<!-- [[Matrix splitting]] links here. Please do not change. -->
{{Main|Linear equation|System of linear equations}}
Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if '''A''' is an ''m''-by-''n'' matrix, '''x''' designates a column vector (that is, ''n''×1-matrix) of ''n'' variables ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>, and '''b''' is an ''m''×1-column vector, then the matrix equation
:<math>\mathbf{Ax} = \mathbf{b}</math>

is equivalent to the system of linear equations<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=I.2.21 and 22}}</ref>
:<math>\begin{align}
  a_{1,1}x_1 + a_{1,2}x_2 + &\cdots + a_{1,n}x_n = b_1 \\
                            &\ \ \vdots \\
  a_{m,1}x_1 + a_{m,2}x_2 + &\cdots + a_{m,n}x_n = b_m
\end{align}</math>

Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If ''n'' = ''m'' and the equations are [[independent equation|independent]], this can be done by writing
:<math>\mathbf{x} = \mathbf{A}^{-1} \mathbf{b}</math>

where '''A'''<sup>−1</sup> is the [[inverse matrix]] of '''A'''. If '''A''' has no inverse, solutions if any can be found using its [[generalized inverse]].

==Linear transformations==
{{Main|Linear transformation|Transformation matrix}}
[[File:Area parallellogram as determinant.svg|thumb|right|The vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram.]]
Matrices and matrix multiplication reveal their essential features when related to ''linear transformations'', also known as ''linear maps''. <span id="linear_maps">A real ''m''-by-''n'' matrix '''A''' gives rise to a linear transformation '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> mapping each vector '''x''' in '''R'''<sup>''n''</sup> to the (matrix) product '''Ax''', which is a vector in '''R'''<sup>''m''</sup>. Conversely, each linear transformation ''f'': '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> arises from a unique ''m''-by-''n'' matrix '''A''': explicitly, the {{nowrap|(''i'', ''j'')-entry}} of '''A''' is the ''i''<sup>th</sup> coordinate of ''f''('''e'''<sub>''j''</sub>), where {{nowrap begin}}'''e'''<sub>''j''</sub> = (0,...,0,1,0,...,0){{nowrap end}} is the [[unit vector]] with 1 in the ''j''<sup>th</sup> position and 0 elsewhere.</span> The matrix '''A''' is said to represent the linear map ''f'', and '''A''' is called the ''transformation matrix'' of ''f''.

For example, the 2×2 matrix
:<math>\mathbf{A} = \begin{bmatrix} a & c\\b & d \end{bmatrix}</math>

can be viewed as the transform of the [[unit square]] into a [[parallelogram]] with vertices at {{nowrap|(0, 0)}}, {{nowrap|(''a'', ''b'')}}, {{nowrap|(''a'' + ''c'', ''b'' + ''d'')}}, and {{nowrap|(''c'', ''d'')}}. The parallelogram pictured at the right is obtained by multiplying '''A''' with each of the column vectors <math>\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix}</math>, and <math>\begin{bmatrix}0 \\ 1\end{bmatrix}</math> in turn. These vectors define the vertices of the unit square.

The following table shows a number of [[2 × 2 real matrices|2-by-2 matrices]] with the associated linear maps of '''R'''<sup>2</sup>. The blue original is mapped to the green grid and shapes. The origin (0,0) is marked with a black point.
{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
|-
| [[Shear mapping|Horizontal shear]] with m = 1.25.
| [[Reflection (mathematics)|Reflection]] through the vertical axis
| [[Squeeze mapping]] with r = 3/2
| [[Scaling (geometry)|Scaling]] by a factor of 3/2
|<span id="rotation_matrix">[[Rotation matrix|Rotation]] by π/6 = 30°</span>
|-
| <math>\begin{bmatrix}
1 & 1.25 \\
0 & 1
\end{bmatrix}</math>
| <math>\begin{bmatrix}
-1 & 0 \\
 0 & 1
\end{bmatrix}</math>
| <math>\begin{bmatrix}
\frac{3}{2} & 0 \\
          0 & \frac{2}{3}
\end{bmatrix}</math>
|<math>\begin{bmatrix}
\frac{3}{2} & 0 \\
          0 & \frac{3}{2}
\end{bmatrix}</math>
|<math>\begin{bmatrix}
\cos\left(\frac{\pi}{6}\right) & -\sin\left(\frac{\pi}{6}\right) \\
\sin\left(\frac{\pi}{6}\right) &  \cos\left(\frac{\pi}{6}\right)
\end{bmatrix}</math>
|-
| width="20%" | [[File:VerticalShear m=1.25.svg|175px]]
| width="20%" | [[File:Flip map.svg|150px]]
| width="20%" | [[File:Squeeze r=1.5.svg|150px]]
| width="20%" | [[File:Scaling by 1.5.svg|125px]]
| width="20%" | [[File:Rotation by pi over 6.svg|125px]]
|}

Under the [[bijection|1-to-1 correspondence]] between matrices and linear maps, matrix multiplication corresponds to [[function composition|composition]] of maps:<ref>{{Harvard citations |last1=Greub |year=1975 |nb=yes |loc=Section III.2}}</ref> if a ''k''-by-''m'' matrix '''B''' represents another linear map ''g'': '''R'''<sup>''m''</sup> → '''R'''<sup>''k''</sup>, then the composition {{nowrap|''g'' ∘ ''f''}} is represented by '''BA''' since
:(''g'' ∘ ''f'')('''x''') = ''g''(''f''('''x''')) = ''g''('''Ax''') = '''B'''('''Ax''') = ('''BA''')'''x'''.

The last equality follows from the above-mentioned associativity of matrix multiplication.

The [[rank of a matrix]] '''A''' is the maximum number of [[linear independence|linearly independent]] row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition II.3.3}}</ref> Equivalently it is the [[Hamel dimension|dimension]] of the [[image (mathematics)|image]] of the linear map represented by '''A'''.<ref>{{Harvard citations |last1=Greub |year=1975 |nb=yes |loc=Section III.1}}</ref> The [[rank–nullity theorem]] states that the dimension of the [[kernel (matrix)|kernel]] of a matrix plus the rank equals the number of columns of the matrix.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem II.3.22}}</ref>

==Square matrix==
{{Main|Square matrix}}
A [[square matrix]] is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order ''n.'' Any two square matrices of the same order can be added and multiplied.
The entries ''a''<sub>''ii''</sub> form the [[main diagonal]] of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix.

===Main types===
:{| class="wikitable" style="float:right; margin:0ex 0ex 2ex 2ex;"
|-
! Name !! Example with ''n'' = 3
|-
| [[Diagonal matrix]] || style="text-align:center;" | <math>
\begin{bmatrix}
a_{11} & 0      & 0 \\
     0 & a_{22} & 0 \\
     0 & 0      & a_{33} \\
\end{bmatrix}
</math>
|-
| [[Lower triangular matrix]] || style="text-align:center;" | <math>
\begin{bmatrix}
a_{11} &      0 & 0 \\
a_{21} & a_{22} & 0 \\
a_{31} & a_{32} & a_{33} \\
\end{bmatrix}
</math>
|-
| [[Upper triangular matrix]] || style="text-align:center;" | <math>
\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
     0 & a_{22} & a_{23} \\
     0 &      0 & a_{33} \\
\end{bmatrix}
</math>
|}

====Diagonal and triangular matrix====
If all entries of '''A''' below the main diagonal are zero, '''A''' is called an ''upper [[triangular matrix]]''. Similarly if all entries of ''A'' above the main diagonal are zero, '''A''' is called a ''lower triangular matrix''. If all entries outside the main diagonal are zero, '''A''' is called a [[diagonal matrix]].

====Identity matrix====
{{Main|Identity matrix}}
The ''identity matrix'' '''I'''<sub>''n''</sub> of size ''n'' is the ''n''-by-''n'' matrix in which all the elements on the [[main diagonal]] are equal to 1 and all other elements are equal to 0, for example,
:<math>
\mathbf{I}_1 = \begin{bmatrix} 1 \end{bmatrix},
\ \mathbf{I}_2 = \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix},
\ \cdots ,
\ \mathbf{I}_n = \begin{bmatrix}
     1 &      0 & \cdots & 0 \\
     0 &      1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
     0 &      0 & \cdots & 1
\end{bmatrix}
</math>
It is a square matrix of order ''n'', and also a special kind of [[diagonal matrix]]. It is called an identity matrix because multiplication with it leaves a matrix unchanged:
:{{nowrap begin}}'''AI'''<sub>''n''</sub> = '''I'''<sub>''m''</sub>'''A''' = '''A'''{{nowrap end}} for any ''m''-by-''n'' matrix '''A'''.

A nonzero scalar multiple of an identity matrix is called a ''scalar'' matrix. If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field.

====Symmetric or skew-symmetric matrix====
A square matrix '''A''' that is equal to its transpose, that is, {{nowrap begin}}'''A''' = '''A'''<sup>T</sup>{{nowrap end}}, is a [[symmetric matrix]]. If instead, '''A''' is equal to the negative of its transpose, that is, {{nowrap begin}}'''A''' = −'''A'''<sup>T</sup>,{{nowrap end}} then '''A''' is a [[skew-symmetric matrix]]. In complex matrices, symmetry is often replaced by the concept of [[Hermitian matrix|Hermitian matrices]], which satisfy '''A'''<sup>∗</sup> = '''A''', where the star or [[asterisk]] denotes the [[conjugate transpose]] of the matrix, that is, the transpose of the [[complex conjugate]] of '''A'''.

By the [[spectral theorem]], real symmetric matrices and complex Hermitian matrices have an [[eigenbasis]]; that is, every vector is expressible as a [[linear combination]] of eigenvectors. In both cases, all eigenvalues are real.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Theorem 2.5.6}}</ref> This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see [[#Infinite matrices|below]].

====Invertible matrix and its inverse====
A square matrix '''A''' is called ''[[invertible matrix|invertible]]'' or ''non-singular'' if there exists a matrix '''B''' such that
:'''AB''' = '''BA''' = '''I'''<sub>''n''</sub> ,<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.2.28}}</ref><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.5.13}}</ref>
where '''I'''<sub>''n''</sub> is the ''n''×''n'' [[identity matrix]] with 1s on the [[main diagonal]] and 0s elsewhere. If '''B''' exists, it is unique and is called the ''[[Invertible matrix|inverse matrix]]'' of '''A''', denoted '''A'''<sup>−1</sup>.

====Definite matrix====
{| class="wikitable" style="float:right; text-align:center; margin:0ex 0ex 2ex 2ex;"
|-
! [[Positive definite matrix]] !! [[Indefinite matrix]]
|-
| <math>\begin{bmatrix}
\frac{1}{4} & 0 \\
          0 & 1 \\
\end{bmatrix}</math>
| <math>\begin{bmatrix}
\frac{1}{4} & 0 \\
          0 & -\frac{1}{4}
\end{bmatrix}</math>
|-
| ''Q''(''x'', ''y'') = 1/4 ''x''<sup>2</sup> + ''y''<sup>2</sup>
| ''Q''(''x'', ''y'') = 1/4 ''x''<sup>2</sup> − 1/4 ''y''<sup>2</sup>
|-
| [[File:Ellipse in coordinate system with semi-axes labelled.svg|150px]] <br>Points such that ''Q''(''x'',''y'')=1 <br> ([[Ellipse]]).
| [[File:Hyperbola2 SVG.svg|150px]] <br> Points such that ''Q''(''x'',''y'')=1 <br> ([[Hyperbola]]).
|}
A symmetric ''n''×''n''-matrix '''A''' is called [[positive-definite matrix|''positive-definite'']] if the associated [[quadratic form]]
:<span id="quadratic forms">''f''{{spaces|hair}}('''x''') = '''x'''<sup>T</sup>'''A&nbsp;x'''</span>

has a positive value for every nonzero vector '''x''' in '''R'''<sup>''n''</sup>. If ''f''{{spaces|hair}}('''x''') only yields negative values then '''A''' is [[definiteness of a matrix#Negative definite|''negative-definite'']]; if ''f'' does produce both negative and positive values then '''A''' is [[definiteness of a matrix#Indefinite|''indefinite'']].<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Chapter 7}}</ref> If the quadratic form ''f'' yields only non-negative values (positive or zero), the symmetric matrix is called ''positive-semidefinite'' (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Theorem 7.2.1}}</ref> The table at the right shows two possibilities for 2-by-2 matrices.

Allowing as input two different vectors instead yields the [[bilinear form]] associated to '''A''':
:''B''<sub>'''A'''</sub> ('''x''', '''y''') = '''x'''<sup>T</sup>'''Ay'''.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Example 4.0.6, p. 169}}</ref>

====Orthogonal matrix====
{{Main|Orthogonal matrix}}
An ''orthogonal matrix'' is a [[#Square matrices|square matrix]] with [[real number|real]] entries whose columns and rows are [[orthogonal]] [[unit vector]]s (that is, [[orthonormality|orthonormal]] vectors). Equivalently, a matrix '''A''' is orthogonal if its [[transpose]] is equal to its [[invertible matrix|inverse]]:
:<math>\mathbf{A}^\mathrm{T}=\mathbf{A}^{-1}, \,</math>
which entails
:<math>\mathbf{A}^\mathrm{T} \mathbf{A} = \mathbf{A} \mathbf{A}^\mathrm{T} = \mathbf{I}_n,</math>
where '''I'''<sub>''n''</sub> is the [[identity matrix]] of size ''n''.

An orthogonal matrix '''A''' is necessarily [[invertible matrix|invertible]] (with inverse {{nowrap|1='''A'''<sup>&minus;1</sup> = '''A'''<sup>T</sup>}}), [[unitary matrix|unitary]] ({{nowrap|1='''A'''<sup>&minus;1</sup> = '''A'''*}}), and [[normal matrix|normal]] ({{nowrap|1='''A'''*'''A''' = '''AA'''*}}). The [[determinant]] of any orthogonal matrix is either {{math|+1}} or {{math|−1}}. A ''special orthogonal matrix'' is an orthogonal matrix with [[determinant]] +1. As a [[linear transformation]], every orthogonal matrix with determinant {{math|+1}} is a pure [[rotation (mathematics)|rotation]] without reflection, i.e., the transformation preserves the orientation of the transformed structure, while every orthogonal matrix with determinant {{math|-1}} reverses the orientation, i.e., is a composition of a pure [[reflection (mathematics)|reflection]] and a (possibly null) rotation. The identity matrices have determinant {{math|1}}, and are pure rotations by an angle zero.

The [[complex number|complex]] analogue of an orthogonal matrix is a [[unitary matrix]].

===Main operations===

====Trace====
The [[trace of a matrix|trace]], tr('''A''') of a square matrix '''A''' is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned [[#non commutative|above]], the trace of the product of two matrices is independent of the order of the factors:
: tr('''AB''') = tr('''BA''').
This is immediate from the definition of matrix multiplication:
:<math>\operatorname{tr}(\mathbf{AB}) = \sum_{i=1}^m \sum_{j=1}^n a_{ij} b_{ji} = \operatorname{tr}(\mathbf{BA}).</math>
It follows that the trace of the product of more than two matrices is independent of [[cyclic permutation]]s of the matrices, however this does not in general apply for arbitrary permutations (for example, tr('''ABC''') ≠ tr('''BAC'''), in general). Also, the trace of a matrix is equal to that of its transpose, that is,
:{{nowrap begin}}tr('''A''') = tr('''A'''<sup>T</sup>){{nowrap end}}.

====Determinant====
{{Main|Determinant}}
[[File:Determinant example.svg|thumb|300px|right|A linear transformation on '''R'''<sup>2</sup> given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the [[orientation (mathematics)|orientation]], since it turns the counterclockwise orientation of the vectors to a clockwise one.]]

The ''determinant'' det('''A''') or |'''A'''| of a square matrix '''A''' is a number encoding certain properties of the matrix. A matrix is invertible [[if and only if]] its determinant is nonzero. Its [[absolute value]] equals the area (in '''R'''<sup>2</sup>) or volume (in '''R'''<sup>3</sup>) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2-by-2 matrices is given by
:<math>\det \begin{bmatrix}a&b\\c&d\end{bmatrix} = ad-bc.</math>
The determinant of 3-by-3 matrices involves 6 terms ([[rule of Sarrus]]). The more lengthy [[Leibniz formula for determinants|Leibniz formula]] generalises these two formulae to all dimensions.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.2.1}}</ref>

The determinant of a product of square matrices equals the product of their determinants:
:{{nowrap begin}}det('''AB''') = det('''A''') · det('''B''').<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem III.2.12}}</ref>{{nowrap end}}
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Corollary III.2.16}}</ref> Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the [[Laplace expansion]] expresses the determinant in terms of [[minor (linear algebra)|minors]], that is, determinants of smaller matrices.<ref>{{Harvard citations |last1=Mirsky |year=1990 |nb=yes |loc=Theorem 1.4.1}}</ref> This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve [[linear system]]s using [[Cramer's rule]], where the division of the determinants of two related square matrices equates to the value of each of the system's variables.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem III.3.18}}</ref>

====Eigenvalues and eigenvectors====
{{Main|Eigenvalue, eigenvector and eigenspace|l1=Eigenvalues and eigenvectors}}
A number λ and a non-zero vector '''v''' satisfying
:'''Av''' = λ'''v'''
are called an ''eigenvalue'' and an ''eigenvector'' of '''A''', respectively.<ref>''Eigen'' means "own" in [[German language|German]] and in [[Dutch language|Dutch]].</ref><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.4.1}}</ref> The number λ is an eigenvalue of an ''n''×''n''-matrix '''A''' if and only if '''A'''−λ'''I'''<sub>''n''</sub> is not invertible, which is [[logical equivalence|equivalent]] to
:<math>\det(\mathsf{A}-\lambda \mathsf{I}) = 0.</math><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.4.9}}</ref>
The polynomial ''p''<sub>'''A'''</sub> in an [[indeterminate (variable)|indeterminate]] ''X'' given by evaluation the determinant det(''X'''''I'''<sub>''n''</sub>−'''A''') is called the [[characteristic polynomial]] of '''A'''. It is a [[monic polynomial]] of [[degree of a polynomial|degree]] ''n''. Therefore the polynomial equation ''p''<sub>'''A'''</sub>(λ)&nbsp;=&nbsp;0 has at most ''n'' different solutions, that is, eigenvalues of the matrix.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Corollary III.4.10}}</ref> They may be complex even if the entries of '''A''' are real. According to the [[Cayley–Hamilton theorem]], {{nowrap begin}}''p''<sub>'''A'''</sub>('''A''') = '''0'''{{nowrap end}}, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the [[zero matrix]].

==Computational aspects==
Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms or iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a [[sequence (mathematics)|sequence]] of vectors '''x'''<sub>''n''</sub> [[limit of a sequence|converging]] to an eigenvector when ''n'' tends to [[infinity]].<ref>{{Harvard citations |last1=Householder |year=1975 |loc=Ch. 7 |nb=yes}}</ref>

To choose the most appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called [[numerical linear algebra]].<ref>{{Harvard citations |last1=Bau III |last2=Trefethen |year=1997 |nb=yes}}</ref> As with other numerical situations, two main aspects are the [[complexity analysis|complexity]] of algorithms and their [[numerical stability]].

Determining the complexity of an algorithm means finding [[upper bound]]s or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, [[matrix multiplication algorithm|multiplication of matrices]]. For example, calculating the matrix product of two ''n''-by-''n'' matrix using the definition given above needs ''n''<sup>3</sup> multiplications, since for any of the ''n''<sup>2</sup> entries of the product, ''n'' multiplications are necessary. The [[Strassen algorithm]] outperforms this "naive" algorithm; it needs only ''n''<sup>2.807</sup> multiplications.<ref>{{Harvard citations |last1=Golub |last2=Van Loan |year=1996 |nb=yes |loc=Algorithm 1.3.1}}</ref> A refined approach also incorporates specific features of the computing devices.

In many practical situations additional information about the matrices involved is known. An important case are [[sparse matrix|sparse matrices]], that is, matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems '''Ax''' = '''b''' for sparse matrices '''A''', such as the [[conjugate gradient method]].<ref>{{Harvard citations |last1=Golub |last2=Van Loan |year=1996 |nb=yes |loc=Chapters 9 and 10, esp. section 10.2}}</ref>

An algorithm is, roughly speaking, numerically stable, if little deviations in the input values do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace expansion (adj('''A''') denotes the [[adjugate matrix]] of '''A''')
:'''A'''<sup>−1</sup> = adj('''A''') / det('''A''')
may lead to significant rounding errors if the determinant of the matrix is very small. The [[matrix norm|norm of a matrix]] can be used to capture the [[condition number|conditioning]] of linear algebraic problems, such as computing a matrix's inverse.<ref>{{Harvard citations |last1=Golub |last2=Van Loan |year=1996 |nb=yes |loc=Chapter 2.3}}</ref>

Although most [[computer language]]s are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the [[HP 9830]] had ROM cartridges to add BASIC commands for matrices. Some computer languages such as [[APL (programming language)|APL]] were designed to manipulate matrices, and [[list of numerical-analysis software|various mathematical programs]] can be used to aid computing with matrices.<ref>For example, [[Mathematica]], see {{Harvard citations |last1=Wolfram |year=2003 |loc=Ch. 3.7 |nb=yes}}</ref>

==Decomposition==
{{Main|Matrix decomposition|Matrix diagonalization|Gaussian elimination|Bareiss algorithm}}
There are several methods to render matrices into a more easily accessible form. They are generally referred to as ''matrix decomposition'' or ''matrix factorization'' techniques. The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.

The [[LU decomposition]] factors matrices as a product of lower ('''L''') and an upper [[triangular matrix|triangular matrices]] ('''U''').<ref>{{Harvard citations |last1=Press |last2=Flannery |last3=Teukolsky |year=1992 |nb=yes}}</ref> Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called [[forward substitution|forward and back substitution]]. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The ''Gaussian elimination'' is a similar algorithm; it transforms any matrix to [[row echelon form]].<ref>{{Harvard citations |last1=Stoer |last2=Bulirsch |year=2002 |nb=yes |loc=Section 4.1}}</ref> Both methods proceed by multiplying the matrix by suitable [[elementary matrix|elementary matrices]], which correspond to [[permutation matrix|permuting rows or columns]] and adding multiples of one row to another row. [[Singular value decomposition]] expresses any matrix '''A''' as a product '''UDV'''<sup>∗</sup>, where '''U''' and '''V''' are [[unitary matrix|unitary matrices]] and '''D''' is a diagonal matrix.

[[File:Jordan blocks.svg|right|thumb|250px|An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks.]]
The [[eigendecomposition]] or ''diagonalization'' expresses '''A''' as a product '''VDV'''<sup>−1</sup>, where '''D''' is a diagonal matrix and '''V''' is a suitable invertible matrix.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Theorem 2.5.4}}</ref> If '''A''' can be written in this form, it is called [[diagonalizable matrix|diagonalizable]]. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into [[Jordan normal form]], that is to say matrices whose only nonzero entries are the eigenvalues λ<sub>1</sub> to λ<sub>n</sub> of '''A''', placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Ch. 3.1, 3.2}}</ref> Given the eigendecomposition, the ''n''<sup>th</sup> power of '''A''' (that is, ''n''-fold iterated matrix multiplication) can be calculated via
:'''A'''<sup>''n''</sup> = ('''VDV'''<sup>−1</sup>)<sup>''n''</sup> = '''VDV'''<sup>−1</sup>'''VDV'''<sup>−1</sup>...'''VDV'''<sup>−1</sup> = '''VD'''<sup>''n''</sup>'''V'''<sup>−1</sup>
and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for '''A''' instead. This can be used to compute the [[matrix exponential]] ''e''<sup>'''A'''</sup>, a need frequently arising in solving [[linear differential equation]]s, [[matrix logarithm]]s and [[square root of a matrix|square roots of matrices]].<ref>{{Harvard citations |last1=Arnold |last2=Cooke |year=1992 |nb=yes |loc=Sections 14.5, 7, 8}}</ref> To avoid numerically [[condition number|ill-conditioned]] situations, further algorithms such as the [[Schur decomposition]] can be employed.<ref>{{Harvard citations |last1=Bronson |year=1989 |nb=yes |loc=Ch. 15}}</ref>

==Abstract algebraic aspects and generalizations==
Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general [[field (mathematics)|fields]] or even [[ring (mathematics)|rings]], while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension are [[tensor]]s, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers.<ref>{{Harvard citations |last1=Coburn |year=1955 |nb=yes |loc=Ch. V}}</ref> Matrices, subject to certain requirements tend to form [[group (mathematics)|groups]] known as matrix groups. Similarly under certain conditions matrices form [[ring (mathematics)|rings]] known as [[matrix ring]]s. Though the product of matrices is not in general commutative yet certain matrices form [[field (mathematics)|fields]] known as [[matrix field]]s.

===Matrices with more general entries===
This article focuses on matrices whose entries are real or [[complex number]]s. <span id="more_general_entries">However, matrices can be considered with much more general types of entries than real or complex numbers.</span> As a first step of generalization, any [[field (mathematics)|field]], that is, a [[set (mathematics)|set]] where [[addition]], [[subtraction]], [[multiplication]], and [[division (mathematics)|division]] operations are defined and well-behaved, may be used instead of '''R''' or '''C''', for example [[rational number]]s or [[finite field]]s. For example, [[coding theory]] makes use of matrices over finite fields. Wherever [[eigenvalue]]s are considered, as these are roots of a polynomial they may exist only in a larger field than that of the entries of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (for example, to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an [[algebraically closed field]], such as '''C''', from the outset.

More generally, matrices with entries in a [[ring (mathematics)|ring]] ''R'' are widely used in mathematics.<ref>{{Harvard citations |last1=Lang |year=2002 |nb=yes |loc=Chapter XIII}}</ref> Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(''n'', ''R'') of all square ''n''-by-''n'' matrices over ''R'' is a ring called [[matrix ring]], isomorphic to the [[endomorphism ring]] of the left ''R''-[[module (mathematics)|module]] ''R''<sup>''n''</sup>.<ref>{{Harvard citations |last1=Lang |year=2002 |nb=yes |loc=XVII.1, p. 643}}</ref> If the ring ''R'' is [[commutative ring|commutative]], that is, its multiplication is commutative, then M(''n'', ''R'') is a unitary noncommutative (unless ''n'' = 1) [[associative algebra]] over ''R''. The [[determinant]] of square matrices over a commutative ring ''R'' can still be defined using the [[Leibniz formula (determinant)|Leibniz formula]]; such a matrix is invertible if and only if its determinant is [[invertible]] in ''R'', generalising the situation over a field ''F'', where every nonzero element is invertible.<ref>{{Harvard citations |last1=Lang |year=2002 |nb=yes |loc=Proposition XIII.4.16}}</ref> Matrices over [[superring]]s are called [[supermatrix|supermatrices]].<ref>{{Harvard citations |last1=Reichl |year=2004 |nb=yes |loc=Section L.2}}</ref>

Matrices do not always have all their entries in the same ring&nbsp;– or even in any ring at all. One special but common case is [[block matrix|block matrices]], which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any [[ring (mathematics)|ring]]; but their sizes must fulfil certain compatibility conditions.

===Relationship to linear maps===
Linear maps '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> are equivalent to ''m''-by-''n'' matrices, as described [[#linear maps|above]]. More generally, any linear map {{nowrap|''f'': ''V'' → ''W''}} between finite-[[Hamel dimension|dimensional]] [[vector space]]s can be described by a matrix '''A''' = (''a''<sub>''ij''</sub>), after choosing [[basis (linear algebra)|bases]] '''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub> of ''V'', and '''w'''<sub>1</sub>, ..., '''w'''<sub>''m''</sub> of ''W'' (so ''n'' is the dimension of ''V'' and ''m'' is the dimension of ''W''), which is such that
:<math>f(\mathbf{v}_j) = \sum_{i=1}^m a_{i,j} \mathbf{w}_i\qquad\mbox{for}\ j=1,\ldots,n.</math>
In other words, column ''j'' of ''A'' expresses the image of '''v'''<sub>''j''</sub> in terms of the basis vectors '''w'''<sub>''i''</sub> of ''W''; thus this relation uniquely determines the entries of the matrix '''A'''. The matrix depends on the choice of the bases: different choices of bases give rise to different, but [[matrix equivalence|equivalent matrices]].<ref>{{Harvard citations |last1=Greub |year=1975 |nb=yes |loc=Section III.3}}</ref> Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix '''A'''<sup>T</sup> describes the [[transpose of a linear map|transpose of the linear map]] given by '''A''', with respect to the [[dual space|dual bases]].<ref>{{Harvard citations |last1=Greub |year=1975 |nb=yes |loc=Section III.3.13}}</ref>

These properties can be restated in a more natural way: the [[category (mathematics)|category]] of all matrices with entries in a field <math>k</math> with multiplication as composition is [[equivalence of categories|equivalent]] to the category of finite dimensional [[vector space]]s and linear maps over this field.

More generally, the set of ''m''×''n'' matrices can be used to represent the ''R''-linear maps between the free modules ''R''<sup>''m''</sup> and ''R''<sup>''n''</sup> for an arbitrary ring ''R'' with unity. When ''n''&nbsp;=&nbsp;''m'' composition of these maps is possible, and this gives rise to the [[matrix ring]] of ''n''×''n'' matrices representing the [[endomorphism ring]] of ''R''<sup>''n''</sup>.

===Matrix groups===
{{Main|Matrix group}}
A [[group (mathematics)|group]] is a mathematical structure consisting of a set of objects together with a [[binary operation]], that is, an operation combining any two objects to a third, subject to certain requirements.<ref>See any standard reference in group.</ref> A group in which the objects are matrices and the group operation is matrix multiplication is called a ''matrix group''.<ref>Additionally, the group must be [[closed subset|closed]] in the general linear group.</ref><ref>{{Harvard citations |last1=Baker |year=2003 |nb=yes |loc=Def. 1.30}}</ref> Since in a group every element must be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the [[general linear group]]s.

Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a [[subgroup]] of (that is, a smaller group contained in) their general linear group, called a [[special linear group]].<ref>{{Harvard citations |last1=Baker |year=2003 |nb=yes |loc=Theorem 1.2}}</ref> [[orthogonal matrix|Orthogonal matrices]], determined by the condition
:'''M'''<sup>T</sup>'''M''' = '''I''',
form the [[orthogonal group]].<ref>{{Harvard citations |last1=Artin |year=1991 |nb=yes |loc=Chapter 4.5}}</ref> Every orthogonal matrix has [[determinant]] 1 or −1. Orthogonal matrices with determinant 1 form a subgroup called ''special orthogonal group''.

Every [[finite group]] is [[isomorphic]] to a matrix group, as one can see by considering the [[regular representation]] of the [[symmetric group]].<ref>{{Harvard citations |last1=Rowen |year=2008 |loc=Example 19.2, p. 198 |nb=yes}}</ref> General groups can be studied using matrix groups, which are comparatively well understood, by means of [[representation theory]].<ref>See any reference in representation theory or [[group representation]].</ref>

===Infinite matrices===
It is also possible to consider matrices with infinitely many rows and/or columns<ref>See the item "Matrix" in {{Harvard citations |editor1-last=Itõ |year=1987 |nb=yes}}</ref> even if, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication, and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general.

If ''R'' is any ring with unity, then the ring of endomorphisms of <math>M=\bigoplus_{i\in I}R</math> as a right ''R'' module is isomorphic to the ring of '''column finite matrices''' <math>\mathbb{CFM}_I(R)</math> whose entries are indexed by <math>I\times I</math>, and whose columns each contain only finitely many nonzero entries. The endomorphisms of ''M'' considered as a left ''R'' module result in an analogous object, the '''row finite matrices''' <math>\mathbb{RFM}_I(R)</math> whose rows each only have finitely many nonzero entries.

If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix '''A''' to describe a linear map ''f'': ''V''→''W'', bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector&nbsp;''v'' of [[coefficient]]s, only finitely many entries ''v''<sub>''i''</sub> are nonzero. Now the columns of '''A''' describe the images by ''f'' of individual basis vectors of ''V'' in the basis of ''W'', which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of ''A'' however: in the product '''A'''·''v'' there are only finitely many nonzero coefficients of ''v'' involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover, this amounts to forming a linear combination of the columns of '''A''' that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns does. Products of two matrices of the given type is well defined (provided that the column-index and row-index sets match), is of the same type, and corresponds to the composition of linear maps.

If ''R'' is a [[normed ring]], then the condition of row or column finiteness can be relaxed. With the norm in place, [[absolutely convergent series]] can be used instead of finite sums. For example, the matrices whose column sums are absolutely convergent sequences form a ring. Analogously, the matrices whose row sums are absolutely convergent series also form a ring.

Infinite matrices can also be used to describe [[Hilbert space#Operators on Hilbert spaces|operators on Hilbert spaces]], where convergence and [[continuous function|continuity]] questions arise, which again results in certain constraints that must be imposed. However, the explicit point of view of matrices tends to obfuscate the matter,<ref>"Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." {{Harvard citations |last1=Halmos |year=1982 |loc=p. 23, Chapter 5 |nb=yes}}</ref> and the abstract and more powerful tools of [[functional analysis]] can be used instead.

===Empty matrices===
An ''empty matrix'' is a matrix in which the number of rows or columns (or both) is zero.<ref>"Empty Matrix: A matrix is empty if either its row or column dimension is zero", [http://www.omatrix.com/manual/glossary.htm Glossary], O-Matrix v6 User Guide</ref><ref>"A matrix having at least one dimension equal to zero is called an empty matrix", [http://www.system.nada.kth.se/unix/software/matlab/Release_14.1/techdoc/matlab_prog/ch_dat29.html MATLAB Data Structures]</ref> Empty matrices help dealing with maps involving the [[zero vector space]]. For example, if ''A'' is a 3-by-0 matrix and ''B'' is a 0-by-3 matrix, then ''AB'' is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space ''V'' to itself, while ''BA'' is a 0-by-0 matrix. There is no common notation for empty matrices, but most [[computer algebra system]]s allow creating and computing with them. The determinant of the 0-by-0 matrix is 1 as follows from regarding the [[empty product]] occurring in the Leibniz formula for the determinant as 1. This value is also consistent with the fact that the identity map from any finite dimensional space to itself has determinant&nbsp;1, a fact that is often used as a part of the characterization of determinants.

==Applications==
There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in [[game theory]] and [[economics]], the [[payoff matrix]] encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.<ref>{{Harvard citations |last1=Fudenberg |last2=Tirole |year=1983 |nb=yes |loc=Section 1.1.1}}</ref> [[Text mining]] and automated [[thesaurus]] compilation makes use of [[document-term matrix|document-term matrices]] such as [[tf-idf]] to track frequencies of certain words in several documents.<ref>{{Harvard citations |last1=Manning |year=1999 |loc=Section 15.3.4 |nb=yes}}</ref>

Complex numbers can be represented by particular real 2-by-2 matrices via
:<math>a + ib \leftrightarrow \begin{bmatrix}
a & -b \\
b &  a \end{bmatrix},</math>
under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of [[absolute value]] 1, as [[#rotation matrix|above]]. A similar interpretation is possible for [[quaternion]]s<ref>{{Harvard citations |last1=Ward |year=1997 |loc=Ch. 2.8 |nb=yes}}</ref> and [[Clifford algebra]]s in general.

Early [[encryption]] techniques such as the [[Hill cipher]] also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.<ref>{{Harvard citations |last1=Stinson |year=2005 |loc=Ch. 1.1.5 and 1.2.4 |nb=yes}}</ref> [[Computer graphics]] uses matrices both to represent objects and to calculate transformations of objects using affine [[rotation matrix|rotation matrices]] to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation.<ref>{{Harvard citations |last1=Association for Computing Machinery |year=1979 |loc=Ch. 7 |nb=yes}}</ref> Matrices over a [[polynomial ring]] are important in the study of [[control theory]].

[[Chemistry]] makes use of matrices in various ways, particularly since the use of [[quantum mechanics|quantum theory]] to discuss [[chemical bond|molecular bonding]] and [[spectroscopy]]. Examples are the [[overlap matrix]] and the [[Fock matrix]] used in solving the [[Roothaan equations]] to obtain the [[molecular orbital]]s of the [[Hartree–Fock method]].

===Graph theory===
[[File:Labelled undirected graph.svg|150px|thumb|right|An undirected graph with adjacency matrix <math>\begin{bmatrix}
1 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{bmatrix}.</math>]]
The [[adjacency matrix]] of a [[finite graph]] is a basic notion of [[graph theory]].<ref>{{Harvard citations |last1=Godsil |last2=Royle |year=2004 |nb=yes |loc=Ch. 8.1}}</ref> It records which vertices of the graph are connected by an edge. Matrices containing just two different values (1 and 0 meaning for example "yes" and "no", respectively) are called [[logical matrix|logical matrices]]. The [[distance matrix|distance (or cost) matrix]] contains information about distances of the edges.<ref>{{Harvard citations |last1=Punnen |year=2002 |nb=yes}}</ref> These concepts can be applied to [[website]]s connected by [[hyperlink]]s or cities connected by roads etc., in which case (unless the connection network is extremely dense) the matrices tend to be [[sparse matrix|sparse]], that is, contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in [[network theory]].

===Analysis and geometry===
The [[Hessian matrix]] of a [[differentiable function]] ''ƒ'': '''R'''<sup>''n''</sup> → '''R''' consists of the [[second derivative]]s of ''ƒ'' with respect to the several coordinate directions, that is,<ref>{{Harvard citations |last1=Lang |year=1987a |nb=yes |loc=Ch. XVI.6}}</ref>
:<math>H(f) = \left [\frac {\partial^2f}{\partial x_i \, \partial x_j} \right ].</math>
[[File:Saddle Point SVG.svg|left|thumb|At the [[saddle point]] (''x''&nbsp;=&nbsp;0, ''y''&nbsp;=&nbsp;0) (red) of the function ''f''(''x'',−''y'') =&nbsp;''x''<sup>2</sup>&nbsp;−&nbsp;''y''<sup>2</sup>, the Hessian matrix <math>\begin{bmatrix}
2 &  0 \\
0 & -2
\end{bmatrix}</math> is [[indefinite matrix|indefinite]].]]It encodes information about the local growth behaviour of the function: given a [[critical point (mathematics)|critical point]] '''x'''&nbsp;=&nbsp;(''x''<sub>1</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>), that is, a point where the first [[partial derivative]]s <math>\partial f / \partial x_i</math> of ''ƒ'' vanish, the function has a [[local minimum]] if the Hessian matrix is [[definiteness of a matrix|positive definite]]. [[Quadratic programming]] can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see [[#quadratic forms|above]]).<ref>{{Harvard citations |last1=Nocedal |year=2006 |loc=Ch. 16 |nb=yes}}</ref>

Another matrix frequently used in geometrical situations is the <span id="Jacobi_matrix">[[Jacobian matrix and determinant|Jacobi matrix]]</span> of a differentiable map ''f'': '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>. If ''f''<sub>1</sub>, ..., ''f''<sub>''m''</sub> denote the components of ''f'', then the Jacobi matrix is defined as<ref>{{Harvard citations |last1=Lang |year=1987a |nb=yes |loc=Ch. XVI.1}}</ref>
:<math>J(f) = \left [\frac {\partial f_i}{\partial x_j} \right ]_{1 \leq i \leq m, 1 \leq j \leq n}.</math>
If ''n'' > ''m'', and if the rank of the Jacobi matrix attains its maximal value ''m'', ''f'' is locally invertible at that point, by the [[implicit function theorem]].<ref>{{Harvard citations |last1=Lang |year=1987a |nb=yes |loc=Ch. XVI.5}}. For a more advanced, and more general statement see {{Harvard citations|last1=Lang|year=1969|nb=yes|loc=Ch. VI.2}}</ref>

[[Partial differential equation]]s can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For [[elliptic partial differential equation]]s this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.<ref>{{Harvard citations |last1=Gilbarg |last2=Trudinger |year=2001 |nb=yes}}</ref>

The [[finite element method]] is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.<ref>{{Harvard citations |last1=Šolin |year=2005 |nb=yes |loc=Ch. 2.5}}. See also [[stiffness method]].</ref>

{{Clear}}
===Probability theory and statistics===
[[File:Markov chain SVG.svg|right|thumb|280px|Two different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2". Both limiting values can be determined from the transition matrices, which are given by <math>
\begin{bmatrix}
 0.7 & 0\\
 0.3 & 1
\end{bmatrix}</math> (red) and <math>
\begin{bmatrix}
 0.7 & 0.2\\
 0.3 & 0.8
\end{bmatrix}</math> (black).]]
[[Stochastic matrix|Stochastic matrices]] are square matrices whose rows are [[probability vector]]s, that is, whose entries are non-negative and sum up to one. Stochastic matrices are used to define [[Markov chain]]s with finitely many states.<ref>{{Harvard citations |last1=Latouche |last2=Ramaswami |year=1999 |nb=yes}}</ref> A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain like [[absorbing state]]s, that is, states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.<ref>{{Harvard citations |last1=Mehata |last2=Srinivasan |year=1978 |nb=yes |loc=Ch. 2.8}}</ref>

Statistics also makes use of matrices in many different forms.<ref>{{Citation |last=Healy |first=Michael |title=Matrices for Statistics |year=1986 |publisher=[[Oxford University Press]] |isbn=978-0-19-850702-4 |authorlink=Michael Healy (statistician)}}</ref> [[Descriptive statistics]] is concerned with describing data sets, which can often be represented as [[data matrix (multivariate statistics)|data matrices]], which may then be subjected to [[dimensionality reduction]] techniques. The [[covariance matrix]] encodes the mutual [[variance]] of several [[random variable]]s.<ref>{{Harvard citations |last1=Krzanowski |year=1988 |loc=Ch. 2.2., p. 60 |nb=yes}}</ref> Another technique using matrices are [[linear least squares]], a method that approximates a finite set of pairs (''x''<sub>1</sub>, ''y''<sub>1</sub>), (''x''<sub>2</sub>, ''y''<sub>2</sub>), ..., (''x''<sub>''N''</sub>, ''y''<sub>''N''</sub>), by a linear function
:''y''<sub>''i''</sub> ≈ ''ax''<sub>''i''</sub> + ''b'', ''i'' = 1, ..., ''N''

which can be formulated in terms of matrices, related to the [[singular value decomposition]] of matrices.<ref>{{Harvard citations |last1=Krzanowski |year=1988 |loc=Ch. 4.1 |nb=yes}}</ref>

[[Random matrix|Random matrices]] are matrices whose entries are random numbers, subject to suitable [[probability distribution]]s, such as [[matrix normal distribution]]. Beyond probability theory, they are applied in domains ranging from [[number theory]] to [[physics]].<ref>{{Harvard citations |authorlink1=Brian Conrey|last1=Conrey |year=2007 |nb=yes}}</ref><ref>{{Harvard citations |last1=Zabrodin |last2=Brezin |last3=Kazakov |last4=Serban |last5=Wiegmann |year=2006 |nb=yes}}</ref>

===Symmetries and transformations in physics===
{{Further|Symmetry in physics}}
Linear transformations and the associated [[symmetry|symmetries]] play a key role in modern physics. For example, [[elementary particle]]s in [[quantum field theory]] are classified as representations of the [[Lorentz group]] of special relativity and, more specifically, by their behavior under the [[spin group]]. Concrete representations involving the [[Pauli matrices]] and more general [[gamma matrices]] are an integral part of the physical description of [[fermion]]s, which behave as [[spinor]]s.<ref>{{Harvard citations |last1=Itzykson |last2=Zuber |year=1980 |nb=yes |loc=Ch. 2}}</ref> For the three lightest [[quark]]s, there is a group-theoretical representation involving the [[special unitary group]] SU(3); for their calculations, physicists use a convenient matrix representation known as the [[Gell-Mann matrices]], which are also used for the SU(3) [[gauge group]] that forms the basis of the modern description of strong nuclear interactions, [[quantum chromodynamics]]. The [[Cabibbo–Kobayashi–Maskawa matrix]], in turn, expresses the fact that the basic quark states that are important for [[weak interaction]]s are not the same as, but linearly related to the basic quark states that define particles with specific and distinct [[mass]]es.<ref>see {{Harvard citations |last1=Burgess |last2=Moore |year=2007 |nb=yes |loc=section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi–Maskawa matrix)}}</ref>

===Linear combinations of quantum states===
The first model of [[quantum mechanics]] ([[Werner Heisenberg|Heisenberg]], 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states.<ref>{{Harvard citations |last1=Schiff |year=1968 |nb=yes |loc=Ch. 6}}</ref> This is also referred to as [[matrix mechanics]]. One particular example is the [[density matrix]] that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" [[eigenstates]].<ref>{{Harvard citations |last1=Bohm |year=2001 |nb=yes |loc=sections II.4 and II.8}}</ref>

Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in [[particle accelerator]]s, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the [[S-matrix]], which encodes all information about the possible interactions between particles.<ref>{{Harvard citations |last1=Weinberg |year=1995 |nb=yes |loc=Ch. 3}}</ref>

===Normal modes===
A general application of matrices in physics is to the description of linearly coupled harmonic systems. The [[equations of motion]] of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system's [[eigenvector]]s, its [[normal mode]]s, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of [[molecules]]: the internal vibrations of systems consisting of mutually bound component atoms.<ref>{{Harvard citations |last1=Wherrett |year=1987 |nb=yes |loc=part II}}</ref> They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.<ref>{{Harvard citations |last1=Riley |last2=Hobson |last3=Bence |year=1997 |nb=yes |loc=7.17}}</ref>

===Geometrical optics===
[[Geometrical optics]] provides further matrix applications. In this approximative theory, the [[light wave|wave nature]] of light is neglected. The result is a model in which [[ray (optics)|light rays]] are indeed [[ray (geometry)|geometrical rays]]. If the deflection of light rays by optical elements is small, the action of a [[lens (optics)|lens]] or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called [[ray transfer matrix analysis]]: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, viz. a ''refraction matrix'' describing the refraction at a lens surface, and a ''translation matrix'', describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies.
The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices.<ref>{{Harvard citations |last1=Guenther |year=1990 |nb=yes |loc=Ch. 5}}</ref>

===Electronics===
Traditional [[mesh analysis]] and [[nodal analysis]] in electronics lead to a system of linear equations that can be described with a matrix.

The behaviour of many [[electronic component]]s can be described using matrices. Let ''A'' be a 2-dimensional vector with the component's input voltage ''v''<sub>1</sub> and input current ''i''<sub>1</sub> as its elements, and let ''B'' be a 2-dimensional vector with the component's output voltage ''v''<sub>2</sub> and output current ''i''<sub>2</sub> as its elements. Then the behaviour of the electronic component can be described by ''B'' = ''H'' '''·''' ''A'', where ''H'' is a 2 x 2 matrix containing one [[electrical impedance|impedance]] element (''h''<sub>12</sub>), one [[admittance]] element (''h''<sub>21</sub>), and two [[dimensionless quantity|dimensionless]] elements (''h''<sub>11</sub> and ''h''<sub>22</sub>). Calculating a circuit now reduces to multiplying matrices.

==History==
Matrices have a long history of application in solving [[linear equation]]s but they were known as arrays until the 1800s. The [[Chinese mathematics|Chinese text]] ''[[The Nine Chapters on the Mathematical Art]]'' written in 10th–2nd century BCE is the first example of the use of array methods to solve [[system of linear equations|simultaneous equations]],<ref>{{Harvard citations |last1=Shen |last2=Crossley |last3=Lun |year=1999 |nb=yes}} cited by {{Harvard citations |last1=Bretscher |year=2005|nb=yes|loc=p. 1}}</ref> including the concept of [[determinant]]s. In 1545 Italian mathematician [[Gerolamo Cardano]] brought the method to Europe when he published ''Ars Magna''.<ref name=":1">''Discrete Mathematics'' 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 {{ISBN|978-0321079121}}, p. 564-565</ref> The [[Japanese mathematics|Japanese mathematician]] [[Seki Kowa|Seki]] used the same array methods to solve simultaneous equations in 1683.<ref>{{cite book |last1=Needham |first1=Joseph |authorlink1=Joseph Needham |last2=Wang Ling |authorlink2=Wang Ling (historian) |title=Science and Civilisation in China |url=https://books.google.com/books?id=jfQ9E0u4pLAC&pg=PA117 |volume=III |year=1959 |publisher=Cambridge University Press |location=Cambridge |isbn=9780521058018 |page=117}}</ref> The Dutch Mathematician'' ''[[Jan de Witt]] represented transformations using arrays in his 1659 book ''Elements of Curves'' (1659).<ref name=":0">''Discrete Mathematics'' 4th Ed. Dossey, Otto, Spense, Vanden Eynden, Published by Addison Wesley, October 10, 2001 {{ISBN|978-0321079121}}, p. 564</ref> Between 1700 and 1710 [[Gottfried Wilhelm Leibniz]] publicized the use of arrays for recording information or solutions and experimented with over 50 different systems of arrays.<ref name=":1"/> [[Gabriel Cramer|Cramer]] presented [[Cramer's rule|his rule]] in 1750.

The term "matrix" (Latin for "womb", derived from ''[[wikt:mater#Latin|mater]]''—mother<ref>{{Citation |url=https://www.merriam-webster.com/dictionary/matrix |title=Merriam-Webster dictionary |accessdate=April 20, 2009 |publisher=Merriam-Webster}}</ref>) was coined by [[James Joseph Sylvester]] in 1850,<ref>Although many sources state that J. J. Sylvester coined the mathematical term "matrix" in 1848, Sylvester published nothing in 1848. (For proof that Sylvester published nothing in 1848, see: J. J. Sylvester with H. F. Baker, ed., ''The Collected Mathematical Papers of James Joseph Sylvester'' (Cambridge, England: Cambridge University Press, 1904), [https://books.google.com/books?id=r-kZAQAAIAAJ&pg=PR6#v=onepage&q&f=false vol. 1.]) His earliest use of the term "matrix" occurs in 1850 in: J. J. Sylvester (1850) "Additions to the articles in the September number of this journal, "On a new class of theorems," and on Pascal's theorem," ''The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', '''37''': 363-370. [https://books.google.com/books?id=CBhDAQAAIAAJ&pg=PA369#v=onepage&q&f=false From page 369]: "For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This does not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants ... "</ref> who understood a matrix as an object giving rise to a number of determinants today called [[minor (linear algebra)|minors]], that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:
:I have in previous papers defined a "Matrix" as a rectangular array of terms, out of which different systems of determinants may be engendered as from the womb of a common parent.<ref>The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, [https://books.google.com/books?id=5GQPlxWrDiEC&pg=PA247 Paper 37], p. 247</ref>

[[Arthur Cayley]] published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done. Instead he defined operations such as addition, subtraction, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held true. Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition.<ref name=":1"/> Early matrix theory had limited the use of arrays almost exclusively to determinants and Arthur Cayley's abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858 [[Arthur Cayley|Cayley]] published his ''A memoir on the theory of matrices''<ref>''Phil.Trans.'' 1858, vol.148, pp.17-37 ''Math. Papers II'' 475-496</ref><ref>{{Harvard citations |editor1-last=Dieudonné |year=1978 |loc=Vol. 1, Ch. III, p. 96 |nb=yes}}</ref> in which he proposed and demonstrated the [[Cayley–Hamilton theorem]].<ref name=":1"/>

An English mathematician named Cullis was the first to use modern bracket notation for matrices in 1913 and he simultaneously demonstrated the first significant use of the notation '''A''' = [''a''<sub>''i'',''j''</sub>] to represent a matrix where ''a''<sub>''i'',''j''</sub> refers to the'' i''th row and the ''j''th column.<ref name=":1"/>

The modern study of determinants sprang from several sources.<ref>{{Harvard citations |last1=Knobloch |year=1994 |nb=yes}}</ref> [[number theory|Number-theoretical]] problems led [[Gauss]] to relate coefficients of [[quadratic form]]s, that is, expressions such as {{nowrap|''x''<sup>2</sup> + ''xy'' − 2''y''<sup>2</sup>,}} and [[linear map]]s in three dimensions to matrices. [[Gotthold Eisenstein|Eisenstein]] further developed these notions, including the remark that, in modern parlance, [[matrix product]]s are [[non-commutative]]. [[Augustin-Louis Cauchy|Cauchy]] was the first to prove general statements about determinants, using as definition of the determinant of a matrix '''A''' = [''a''<sub>''i'',''j''</sub>] the following: replace the powers ''a''<sub>''j''</sub><sup>''k''</sup> by ''a''<sub>''jk''</sub> in the [[polynomial]]
:<math>a_1 a_2 \cdots a_n \prod_{i < j} (a_j - a_i)\;</math>,
where Π denotes the [[multiplication|product]] of the indicated terms. He also showed, in 1829, that the [[eigenvalue]]s of symmetric matrices are real.<ref>{{Harvard citations |last1=Hawkins |year=1975 |nb=yes}}</ref> [[Carl Gustav Jacob Jacobi|Jacobi]] studied "functional determinants"—later called [[Jacobian matrix and determinant|Jacobi determinant]]s by Sylvester—which can be used to describe geometric transformations at a local (or [[infinitesimal]]) level, see [[#Jacobi matrix|above]]; [[Leopold Kronecker|Kronecker]]'s ''Vorlesungen über die Theorie der Determinanten''<ref>{{Harvard citations |last1=Kronecker |editor1-last=Hensel |year=1897 |nb=yes}}</ref> and [[Karl Weierstrass|Weierstrass']] ''Zur Determinantentheorie'',<ref>{{Harvard citations |last1=Weierstrass |year=1915 |volume=3 |loc=pp. 271–286 |nb=yes}}</ref> both published in 1903, first treated determinants [[axiom]]atically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.

Many theorems were first established for small matrices only, for example the [[Cayley–Hamilton theorem]] was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by [[William Rowan Hamilton|Hamilton]] for 4×4 matrices. [[Georg Frobenius|Frobenius]], working on [[bilinear form]]s, generalized the theorem to all dimensions (1898). Also at the end of the 19th century the [[Gauss–Jordan elimination]] (generalizing a special case now known as [[Gauss elimination]]) was established by [[Wilhelm Jordan (geodesist)|Jordan]]. In the early 20th century, matrices attained a central role in linear algebra,<ref>{{Harvard citations |last1=Bôcher |year=2004 |nb=yes}}</ref> partially due to their use in classification of the [[hypercomplex number]] systems of the previous century.

The inception of [[matrix mechanics]] by [[Werner Heisenberg|Heisenberg]], [[Max Born|Born]] and [[Pascual Jordan|Jordan]] led to studying matrices with infinitely many rows and columns.<ref>{{Harvard citations |last1=Mehra |last2=Rechenberg |year=1987 |nb=yes}}</ref> Later, [[John von Neumann|von Neumann]] carried out the [[mathematical formulation of quantum mechanics]], by further developing [[functional analysis|functional analytic]] notions such as [[linear operator]]s on [[Hilbert space]]s, which, very roughly speaking, correspond to [[Euclidean space]], but with an infinity of [[Hamel dimension|independent directions]].

===Other historical usages of the word "matrix" in mathematics===
The word has been used in unusual ways by at least two authors of historical importance.

[[Bertrand Russell]] and [[Alfred North Whitehead]] in their ''Principia Mathematica'' (1910–1913) use the word "matrix" in the context of their [[axiom of reducibility]]. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the "bottom" (0 order) the function is identical to its [[extension (predicate logic)|extension]]:
:"Let us give the name of ''matrix'' to any function, of however many variables, that does not involve any [[apparent variable]]s. Then, any possible function other than a matrix derives from a matrix by means of generalization, that is, by considering the proposition that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined".<ref>Whitehead, Alfred North; and Russell, Bertrand (1913) ''Principia Mathematica to *56'', Cambridge at the University Press, Cambridge UK (republished 1962) cf page 162ff.</ref>
For example, a function Φ(''x, y'') of two variables ''x'' and ''y'' can be reduced to a ''collection'' of functions of a single variable, for example, ''y'', by "considering" the function for all possible values of "individuals" ''a<sub>i</sub>'' substituted in place of variable ''x''. And then the resulting collection of functions of the single variable ''y'', that is, ∀a<sub>i</sub>: Φ(''a<sub>i</sub>, y''), can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" ''b<sub>i</sub>'' substituted in place of variable ''y'':
:∀b<sub>j</sub>∀a<sub>i</sub>: Φ(''a<sub>i</sub>, ''b<sub>j</sub>'').''

[[Alfred Tarski]] in his 1946 ''Introduction to Logic'' used the word "matrix" synonymously with the notion of [[truth table]] as used in mathematical logic.<ref>Tarski, Alfred; (1946) ''Introduction to Logic and the Methodology of Deductive Sciences'', Dover Publications, Inc, New York NY, {{ISBN|0-486-28462-X}}.</ref>

==See also==
{{Portal|Mathematics}}
{{Div col|colwidth=30em}}

* [[Algebraic multiplicity]]
* [[Geometric multiplicity]]
* [[Gram–Schmidt process]]
* [[List of matrices]]
* [[Matrix calculus]]
* [[Matrix function]]
* [[Periodic matrix set]]
* [[Tensor]]

{{div col end}}

==Notes==
{{Reflist|colwidth=30em}}
{{Reflist|group=nb|3}}

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==External links==
{{Commons category}}
{{Wikibooks|Linear Algebra|Matrices}}
{{Wikiversity|at=Linear algebra#Matrices|Matrices}}
; Encyclopedic articles
* {{SpringerEOM|title=Matrix|id=p/m062780}}

; History
* [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_determinants.html MacTutor: Matrices and determinants]
* [http://www.economics.soton.ac.uk/staff/aldrich/matrices.htm Matrices and Linear Algebra on the Earliest Uses Pages]
* [http://jeff560.tripod.com/matrices.html Earliest Uses of Symbols for Matrices and Vectors]

; Online books
* {{Citation |last1=Kaw |first1=Autar K. |title=Introduction to Matrix Algebra |url=http://autarkaw.com/books/matrixalgebra/index.html |isbn=978-0-615-25126-4}}
* {{Citation |title=The Matrix Cookbook |url=http://www.math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf |accessdate=24 March 2014 |format=PDF}}
* {{Citation |last1=Brookes |first1=Mike |title=The Matrix Reference Manual |url=http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html |publisher=[[Imperial College]] |location=London |year=2005 |accessdate=10 Dec 2008}}

; Online matrix calculators
* {{Citation |title=matrixcalc (Matrix Calculator) |url=https://matrixcalc.org/en/}}
* {{Citation |title=SimplyMath (Matrix Calculator) |url=https://simplemath.online/linear-algebra/matrix-and-vector.html}}
* {{Citation |title=Free C++ Library|url=https://github.com/nom-de-guerre/Matrices}}
* {{Citation |title=Matrix Calculator (DotNumerics) |url=http://www.dotnumerics.com/MatrixCalculator/}}
* {{Citation |last1=Xiao |first1=Gang |title=Matrix calculator |url=http://wims.unice.fr/wims/wims.cgi?module=tool/linear/matrix.en |accessdate=10 Dec 2008}}
* {{Citation |title=Online matrix calculator |url=http://www.bluebit.gr/matrix-calculator/ |accessdate=10 Dec 2008}}
* {{Citation |title=Online matrix calculator (ZK framework) |url=http://matrixcalc.info/MatrixZK/ |accessdate=26 Nov 2009 |url-status=unfit |archiveurl=https://web.archive.org/web/20130512101418/http://matrixcalc.info/MatrixZK/ |archivedate=2013-05-12}}
* {{Citation |title=MacAnova |url=http://www.stat.umn.edu/macanova/macanova.home.html |last1=Oehlert |first1=Gary W. |last2=Bingham |first2=Christopher |publisher=[[University of Minnesota]], School of Statistics |accessdate=10 Dec 2008}}, a freeware package for matrix algebra and statistics
* {{Citation |title=Online matrix calculator |url=https://www.idomaths.com/matrix.php |accessdate=14 Dec 2009}}
* [http://www.elektro-energetika.cz/calculations/matreg.php?language=english Operation with matrices in R (determinant, track, inverse, adjoint, transpose)]
* [https://www.wolframalpha.com/widgets/view.jsp?id=7edcfb2d8f6a659ef4cd1e6c9b6d7079 Matrix operations widget in Wolfram|Alpha]

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