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{{Use American English|date = March 2019}}
{{Short description|Matrix whose only nonzero elements are on its main diagonal}}
In [[linear algebra]], a '''diagonal matrix''' is a [[matrix (mathematics)|matrix]] in which the entries outside the [[main diagonal]] are all zero; the term usually refers to [[square matrices]]. An example of a 2-by-2 diagonal matrix is <math>\left[\begin{smallmatrix}
3 & 0 \\
0 & 2 \end{smallmatrix}\right]</math>, while an example of a 3-by-3 diagonal matrix is<math>
\left[\begin{smallmatrix}
6 & 0 & 0 \\
0 & 7 & 0 \\
0 & 0 & 4
\end{smallmatrix}\right]</math>. An [[identity matrix]] of any size, or any multiple of it (a [[#Scalar matrix|scalar matrix]]), will be a diagonal matrix.

==Definition==

As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero.  That is, the matrix {{nowrap|1=''D'' = (''d''<sub>''i'',''j''</sub>)}} with ''n'' columns and ''n'' rows is diagonal if

:<math>\forall i,j \in \{1, 2, \ldots, n\}, i \ne j \implies d_{i,j} = 0</math>.

However, the main diagonal entries are unrestricted.

The term ''diagonal matrix'' may sometimes refer to a '''{{visible anchor|rectangular diagonal matrix}}''', which is an ''m''-by-''n'' matrix with all the entries not of the form ''d''<sub>''i'',''i''</sub> being zero. For example:
:<math>\begin{bmatrix}
1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & -3\\
0 & 0 & 0\\
\end{bmatrix}</math> or <math>\begin{bmatrix}
1 & 0 & 0 & 0 & 0\\
0 & 4 & 0& 0 & 0\\
0 & 0 & -3& 0 & 0\end{bmatrix}</math>

More often, however, ''diagonal matrix'' refers to square matrices, which can be specified explicitly as a '''{{visible anchor|square diagonal matrix}}'''. A square diagonal matrix is a [[symmetric matrix]], so this can also be called a '''{{visible anchor|symmetric diagonal matrix}}'''.

The following matrix is square diagonal matrix:
:<math>\begin{bmatrix}
1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & -2\end{bmatrix}</math>

If the entries are [[real numbers]] or [[complex numbers]], then it is a [[normal matrix]] as well.

In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices".

== Scalar matrix ==
<!-- Linked from [[Scalar matrix]] and [[Scalar transformation]] -->
A diagonal matrix with all its main diagonal entries equal is a '''scalar matrix''', that is, a scalar multiple ''&lambda;I'' of the [[identity matrix]] ''I''. Its effect on a [[vector (mathematics and physics)|vector]] is [[scalar multiplication]] by ''&lambda;''. For example, a 3&times;3 scalar matrix has the form:
:<math>
  \begin{bmatrix}
    \lambda &       0 & 0       \\
          0 & \lambda & 0       \\
          0 &       0 & \lambda
  \end{bmatrix} \equiv \lambda \boldsymbol{I}_3
</math>

The scalar matrices are the [[center of an algebra|center]] of the algebra of matrices: that is, they are precisely the matrices that [[commute (mathematics)|commute]] with all other square matrices of the same size.{{efn|Proof: given the [[elementary matrix]] <math>e_{ij}</math>, <math>Me_{ij}</math> is the matrix with only the ''i''-th row of ''M'' and <math>e_{ij}M</math> is the square matrix with only the ''M'' ''j''-th column, so the non-diagonal entries must be zero, and the ''i''th diagonal entry much equal the ''j''th diagonal entry.}} By contrast, over a [[Field (mathematics)|field]] (like the real numbers), a diagonal matrix with all diagonal elements distinct only commutes with diagonal matrices (its [[centralizer]] is the set of diagonal matrices). That is because if a diagonal matrix <math>D = \mathrm{diag}(a_1, \dots, a_n)</math> has <math>a_i != a_j</math>, then given a matrix <math>M</math> with <math>m_{ij} != 0</math>, the <math>(i, j)</math> term of the products are: <math>(DM)_{ij} = a_jm_{ij}</math> and <math>(MD)_{ij} = m_{ij}a_i</math>, and <math>a_jm_{ij} != m_{ij}a_i</math> (since you can divide by <math>m_{ij}</math>), so they do not commute unless the off-diagonal terms are zero.{{efn|Over more general rings, this does not hold, because one cannot always divide.}} Diagonal matrices where the diagonal entries are not all equal or all distinct have centralizers intermediate between the whole space and only diagonal matrices.<ref>{{cite web |url=https://math.stackexchange.com/q/1697991 |title=Do Diagonal Matrices Always Commute? |author=<!--Not stated--> |date=March 15, 2016 |publisher=Stack Exchange |access-date=August 4, 2018 }}</ref>

For an abstract vector space ''V'' (rather than the concrete vector space <math>K^n</math>), or more generally a [[module (ring theory)|module]] ''M'' over a [[ring (algebra)|ring]] ''R'', with the [[endomorphism algebra]] End(''M'') (algebra of linear operators on ''M'') replacing the algebra of matrices, the analog of scalar matrices are '''scalar transformations'''. Formally, scalar multiplication is a linear map, inducing a map <math>R \to \operatorname{End}(M),</math> (send a scalar ''&lambda;'' to the corresponding scalar transformation, multiplication by ''&lambda;'') exhibiting End(''M'') as a ''R''-[[Algebra (ring theory)|algebra]]. For vector spaces, or more generally [[free module]]s <math>M \cong R^n</math>, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the [[center of a ring|center]] of the endomorphism algebra, and similarly invertible transforms are the center of the [[general linear group]] GL(''V''), where they are denoted by Z(''V''), follow the usual notation for the center.

== Vector operations ==
Multiplying a vector by a diagonal matrix multiplies each of the terms by the corresponding diagonal entry. Given a diagonal matrix <math>D = \mathrm{diag}(a_1, \dots, a_n)</math> and a vector <math>v = \left[\begin{smallmatrix}x_1 \\ \vdots \\ x_n\end{smallmatrix}\right]</math>, the product is:
:<math>Dv = \mathrm{diag}(a_1, \dots, a_n)\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix}
= \begin{bmatrix}
a_1 \\
   & \ddots\\
   &        & a_n
\end{bmatrix}
\begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix}
= \begin{bmatrix}a_1x_1 \\ \vdots \\ a_nx_n\end{bmatrix}.</math>

This can be expressed more compactly by using a vector instead of a diagonal matrix, <math>d = \left[\begin{smallmatrix}a_1 \\ \vdots \\ a_n\end{smallmatrix}\right]</math>, and taking the [[Hadamard product (matrices)|Hadamard product]] of the vectors (entrywise product), denoted <math>d \odot v</math>:

:<math>Dv = d \odot v = \begin{bmatrix}a_1 \\ \vdots \\ a_n\end{bmatrix} \odot \begin{bmatrix}x_1 \\ \vdots \\ x_n\end{bmatrix} = \begin{bmatrix}a_1x_1 \\ \vdots \\ a_nx_n\end{bmatrix}.</math>

This is mathematically equivalent, but avoids storing all the zero terms of this [[sparse matrix]]. This product is thus used in computational mathematics, such as computing products of derivatives in [[backpropagation]], since is these cases the entire diagonal matrix is not stored.

== Matrix operations ==
The operations of matrix addition and [[matrix multiplication]] are especially simple for diagonal matrices. Write {{nowrap|diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} for a diagonal matrix whose diagonal entries starting in the upper left corner are ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>. Then, for addition, we have

:{{nowrap|diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} + {{nowrap|diag(''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>)}} = {{nowrap|diag(''a''<sub>1</sub> + ''b''<sub>1</sub>, ..., ''a''<sub>''n''</sub> + ''b''<sub>''n''</sub>)}}

and for [[matrix multiplication]],

:{{nowrap|diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} &middot; {{nowrap|diag(''b''<sub>1</sub>, ..., ''b''<sub>''n''</sub>)}} = {{nowrap|diag(''a''<sub>1</sub>''b''<sub>1</sub>, ..., ''a''<sub>''n''</sub>''b''<sub>''n''</sub>)}}.

The diagonal matrix {{nowrap|diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} is [[invertible matrix|invertible]] [[if and only if]] the entries ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> are all non-zero. In this case, we have

:{{nowrap|diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)<sup>−1</sup>}} = {{nowrap|diag(''a''<sub>1</sub><sup>−1</sup>, ..., ''a''<sub>''n''</sub><sup>−1</sup>)}}.

In particular, the diagonal matrices form a [[subring]] of the ring of all ''n''-by-''n'' matrices.

Multiplying an ''n''-by-''n'' matrix ''A'' from the ''left'' with {{nowrap|diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} amounts to multiplying the ''i''th ''row'' of ''A'' by ''a''<sub>''i''</sub> for all ''i''; multiplying the matrix ''A'' from the ''right'' with {{nowrap|diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} amounts to multiplying the ''i''th ''column'' of ''A'' by ''a''<sub>''i''</sub> for all ''i''.

== Operator matrix in eigenbasis ==

{{Main|Transformation_matrix#Finding the matrix of a transformation|Eigenvalues and eigenvectors|l1=Finding the matrix of a transformation}}

As explained in [[transformation matrix#Finding the matrix of a transformation|determining coefficients of operator matrix]], there is a special basis, ''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>, for which the matrix <math>A</math> takes the diagonal form. Hence, in the defining equation <math>A \vec e_j = \sum a_{i,j} \vec e_i</math>, all coefficients <math>a_{i,j} </math> with ''i'' ≠ ''j'' are zero, leaving only one term per sum. The surviving diagonal elements, <math>a_{i,i}</math>, are known as '''eigenvalues''' and designated with <math>\lambda_i</math> in the equation, which reduces to <math>A \vec e_i = \lambda_i \vec e_i</math>. The resulting equation is known as '''eigenvalue equation'''<ref>{{cite book |last=Nearing |first=James |year=2010 |title=Mathematical Tools for Physics |url=http://www.physics.miami.edu/nearing/mathmethods |chapter=Chapter 7.9: Eigenvalues and Eigenvectors |chapterurl= http://www.physics.miami.edu/~nearing/mathmethods/operators.pdf |accessdate=January 1, 2012|isbn=048648212X}}</ref> and used to derive the [[characteristic polynomial]] and, further, [[eigenvalues and eigenvectors]].

In other words, the [[eigenvalue]]s of {{nowrap|diag(''λ''<sub>1</sub>, ..., ''λ''<sub>''n''</sub>)}} are ''λ''<sub>1</sub>, ..., ''λ''<sub>''n''</sub> with associated [[eigenvectors]] of ''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>.

== Properties ==

The [[determinant]] of {{nowrap|diag(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} is the product ''a''<sub>1</sub>...''a''<sub>''n''</sub>.

The [[adjugate]] of a diagonal matrix is again diagonal.

A square matrix is diagonal if and only if it is triangular and [[Normal matrix|normal]].

Any square diagonal matrix is also a [[symmetric matrix]].

A symmetric diagonal matrix can be defined as a matrix that is both [[triangular matrix|upper-]] and [[triangular matrix|lower-triangular]]. The [[identity matrix]] ''I''<sub>''n''</sub> and any square [[zero matrix]] are diagonal. A one-dimensional matrix is always diagonal.

== Applications ==
Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is typically desirable to represent a given matrix or [[linear operator|linear map]] by a diagonal matrix.

In fact, a given ''n''-by-''n'' matrix ''A'' is [[similar matrix|similar]] to a diagonal matrix (meaning that there is a matrix ''X'' such that ''X''<sup>−1</sup>''AX'' is diagonal) if and only if it has ''n'' [[linearly independent]] eigenvectors. Such matrices are said to be [[diagonalizable matrix|diagonalizable]].

Over the [[field (mathematics)|field]] of [[real number|real]] or [[complex number|complex]] numbers, more is true. The [[spectral theorem]] says that every [[normal matrix]] is [[matrix similarity|unitarily similar]] to a diagonal matrix (if ''AA''<sup>∗</sup> = ''A''<sup>∗</sup>''A'' then there exists a [[unitary matrix]] ''U'' such that ''UAU''<sup>∗</sup> is diagonal). Furthermore, the [[singular value decomposition]] implies that for any matrix ''A'', there exist unitary matrices ''U'' and ''V'' such that ''UAV''<sup>∗</sup> is diagonal with positive entries.

== Operator theory ==
In [[operator theory]], particularly the study of [[PDEs]], operators are particularly easy to understand and PDEs easy to solve if the operator is diagonal with respect to the basis with which one is working; this corresponds to a [[separable partial differential equation]]. Therefore, a key technique to understanding operators is a change of coordinates—in the language of operators, an [[integral transform]]—which changes the basis to an [[eigenbasis]] of [[eigenfunction]]s: which makes the equation separable. An important example of this is the [[Fourier transform]], which diagonalizes constant coefficient differentiation operators (or more generally translation invariant operators), such as the Laplacian operator, say, in the [[heat equation]].

Especially easy are [[multiplication operator]]s, which are defined as multiplication by (the values of) a fixed function–the values of the function at each point correspond to the diagonal entries of a matrix.

== See also ==
{{colbegin}}
* [[Anti-diagonal matrix]]
* [[Banded matrix]]
* [[Bidiagonal matrix]]
* [[Diagonally dominant matrix]]
* [[Jordan normal form]]
* [[Multiplication operator]]
* [[Tridiagonal matrix]]
* [[Toeplitz matrix]]
* [[Toral Lie algebra]]
* [[Circulant matrix]]
{{colend}}

== Notes ==
{{notelist}}

== References ==
{{reflist}}
*{{cite book |first=Roger A. |last=Horn |first2=Charles R. |last2=Johnson |title=Matrix Analysis |publisher=Cambridge University Press |year=1985 |isbn=0-521-30586-1 }}

{{Matrix classes}}

[[Category:Matrix normal forms]]
[[Category:Sparse matrices]]