View source for Symmetric matrix

{{Use American English|date=January 2019}}{{Short description|Matrix equal to its transpose
}}
{{for|matrices with symmetry over the [[complex number]] field|Hermitian matrix}}

[[File:Matrix symmetry qtl1.svg|thumb|Symmetry of a 5×5 matrix]]

In [[linear algebra]], a '''symmetric matrix''' is a [[square matrix]] that is equal to its [[transpose]]. Formally,

{{Equation box 1
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|equation = <math>A \text{ is symmetric} \iff A = A^\textsf{T}.</math>
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Because equal matrices have equal dimensions, only square matrices can be symmetric.

The entries of a symmetric matrix are symmetric with respect to the [[main diagonal]]. So if <math>a_{ij}</math> denotes the entry in the <math>i</math>-th row and <math>j</math>-th column then

{{Equation box 1
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|equation = <math>A \text{ is symmetric} \iff \text{ for every }i,j,\quad a_{ji} = a_{ij}</math>
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for all indices <math>i</math> and <math>j.</math>

Every square [[diagonal matrix]] is 
symmetric, since all off-diagonal elements are zero. Similarly in [[characteristic (algebra)|characteristic]] different from 2, each diagonal element of a [[skew-symmetric matrix]] must be zero, since each is its own negative.

In linear algebra, a [[real number|real]] symmetric matrix represents a [[self-adjoint operator]]<ref>{{Cite book|author=Jesús Rojo García|title=Álgebra lineal |language= Spanish|edition=2nd|publisher=Editorial AC|year=1986|isbn=84-7288-120-2}}</ref> over a [[real number|real]] [[inner product space]]. The corresponding object for a [[complex number|complex]] inner product space is a [[Hermitian matrix]] with complex-valued entries, which is equal to its [[conjugate transpose]]. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries.  Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.

== Example ==
The following <math>3 \times 3</math> matrix is symmetric:

:<math>A =
  \begin{bmatrix}
    1 &  7 &  3 \\
    7 &  4 & -5 \\
    3 & -5 &  6
  \end{bmatrix}
</math>

== Properties ==
===Basic properties===
* The sum and difference of two symmetric matrices is again symmetric

* This is not always true for the [[matrix multiplication|product]]: given symmetric matrices <math>A</math> and <math>B</math>, then <math>AB</math> is symmetric if and only if <math>A</math> and <math>B</math> [[commutativity|commute]], i.e., if <math>AB=BA</math>.

* For integer <math>n</math>, <math>A^n</math> is symmetric if <math>A</math> is symmetric. 

* If <math>A^{-1}</math> exists, it is symmetric if and only if <math>A</math> is symmetric.

===Decomposition into symmetric and skew-symmetric===
Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition.
Let <math>\mbox{Mat}_n</math> denote the space of <math>n \times n</math> matrices. If <math>\mbox{Sym}_n</math> denotes the space of <math>n \times n</math> symmetric matrices and <math>\mbox{Skew}_n</math> the space of <math>n \times n</math> skew-symmetric matrices then <math>\mbox{Mat}_n = \mbox{Sym}_n + \mbox{Skew}_n</math> and <math>\mbox{Sym}_n \cap \mbox{Skew}_n = \{0\}</math>, i.e.
:<math>\mbox{Mat}_n = \mbox{Sym}_n \oplus \mbox{Skew}_n , </math>

where <math>\oplus</math> denotes the [[direct sum of modules|direct sum]]. Let <math>X \in \mbox{Mat}_n</math> then
:<math>X = \frac{1}{2}\left(X + X^\textsf{T}\right) + \frac{1}{2}\left(X - X^\textsf{T}\right)</math>.

Notice that <math>\frac{1}{2}\left(X + X^\textsf{T}\right) \in \mbox{Sym}_n</math> and <math>\frac{1}{2}\left(X - X^\textsf{T}\right) \in \mbox{Skew}_n</math>. This is true for every [[square matrix]] <math>X</math> with entries from any [[field (mathematics)|field]] whose [[characteristic (algebra)|characteristic]] is different from 2.

A symmetric <math>n \times n</math> matrix is determined by <math>\tfrac{1}{2}n(n+1)</math> scalars (the number of entries on or above the [[main diagonal]]). Similarly, a [[skew-symmetric matrix]] is determined by <math>\tfrac{1}{2}n(n-1)</math> scalars (the number of entries above the main diagonal).

=== Matrix congruent to a symmetric matrix ===
Any matrix [[matrix congruence|congruent]] to a symmetric matrix is again symmetric: if <math>X</math> is a symmetric matrix then so is <math>A X A^{\mathrm T}</math> for any matrix <math>A</math>.

=== Symmetry implies normality ===
A (real-valued) symmetric matrix is necessarily a [[normal matrix]].
all eigen values of real symmetric matrix are always real and equal to the order of the any given matrix.

=== Real symmetric matrices ===
<!--If A is a skew-symmetric matrix, then ''iA'' (where ''i'' is an [[imaginary unit]]) is symmetric.-->
Denote by <math>\langle \cdot,\cdot \rangle</math> the standard [[inner product]] on <math>\mathbb{R}^n</math>. The real <math>n \times n</math> matrix <math>A</math> is symmetric if and only if

:<math>\langle Ax, y \rangle = \langle x, Ay \rangle \quad \forall x, y \in \mathbb{R}^n.</math>

Since this definition is independent of the choice of [[basis (linear algebra)|basis]], symmetry is a property that depends only on the [[linear operator]] A and a choice of [[inner product]]. This characterization of symmetry is useful, for example, in [[differential geometry]], for each [[tangent space]] to a [[manifold]] may be endowed with an inner product, giving rise to what is called a [[Riemannian manifold]]. Another area where this formulation is used is in [[Hilbert space]]s.

The finite-dimensional [[spectral theorem]] says that any symmetric matrix whose entries are [[real number|real]] can be [[diagonal matrix|diagonalized]] by an [[orthogonal matrix]]. More explicitly: For every symmetric real matrix <math>A</math> there exists a real orthogonal matrix <math>Q</math> such that <math>D = Q^{\mathrm T} A Q</math> is a [[diagonal matrix]]. Every symmetric matrix is thus, [[up to]] choice of an [[orthonormal basis]], a diagonal matrix.

If <math>A</math> and <math>B</math> are <math>n \times n</math> real symmetric matrices that commute, then they can be simultaneously diagonalized: there exists a basis of <math>\mathbb{R}^n</math> such that every element of the basis is an [[eigenvector]] for both <math>A</math> and <math>B</math>.

Every real symmetric matrix is [[Hermitian matrix|Hermitian]], and therefore all its [[eigenvalues]] are real. (In fact, the eigenvalues are the entries in the diagonal matrix <math>D</math> (above), and therefore <math>D</math> is uniquely determined by <math>A</math> up to the order of its entries.) Essentially, the property of being symmetric for real matrices corresponds to the property of being Hermitian for complex matrices.

=== Complex symmetric matrices ===
A complex symmetric matrix can be 'diagonalized' using a [[unitary matrix]]: thus if <math>A</math> is a complex symmetric matrix, there is a unitary matrix <math>U</math> such that  <math>U A U^{\mathrm T}</math> is a real diagonal matrix with non-negative entries. This result is referred to as the '''Autonne–Takagi factorization'''. It was originally proved by [[Léon Autonne]] (1915) and [[Teiji Takagi]] (1925) and rediscovered with different proofs by several other mathematicians.<ref>{{cite book|first1=R.A.|last1=Horn|first2=C.R.|last2=Johnson|title=Matrix analysis |year=2013 | edition=2nd | publisher = Cambridge University Press | mr = 2978290|at=pp. 263, 278}}</ref><ref>See:
*{{citation|first=L.|last= Autonne|title= Sur les matrices hypohermitiennes et sur les matrices unitaires|journal= Ann. Univ. Lyon|volume= 38|year=1915|pages= 1–77}}
*{{citation|first=T.|last= Takagi|title= On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau|journal= Japan. J. Math.|volume= 1 |year=1925|pages= 83–93|doi= 10.4099/jjm1924.1.0_83}}
*{{citation|title=Symplectic Geometry|first=Carl Ludwig|last= Siegel|journal= American Journal of Mathematics|volume= 65|issue=1|year=1943|pages=1–86|jstor= 2371774|doi=10.2307/2371774|id=Lemma 1, page 12}}
*{{citation|first=L.-K.|last= Hua|title= On the theory of automorphic functions of a matrix variable I–geometric basis|journal= Amer. J. Math.|volume= 66 |issue= 3|year=1944|pages= 470–488|doi=10.2307/2371910|jstor= 2371910}}
*{{citation|first=I.|last= Schur|title= Ein Satz über quadratische formen mit komplexen koeffizienten|journal=Amer. J. Math.|volume=67|issue= 4|year=1945|pages=472–480|doi=10.2307/2371974|jstor= 2371974}}
*{{citation|first1=R.|last1= Benedetti|first2=P.|last2= Cragnolini|title=On simultaneous diagonalization of one Hermitian and one symmetric form|journal= Linear Algebra Appl. |volume=57 |year=1984| pages=215–226|doi=10.1016/0024-3795(84)90189-7}}
</ref> In fact, the matrix <math>B=A^{\dagger} A</math> is Hermitian and [[Definiteness of a matrix|positive semi-definite]], so there is a unitary matrix <math>V</math> such that <math>V^{\dagger} B V</math> is diagonal with non-negative real entries. Thus <math>C=V^{\mathrm T} A V</math> is complex symmetric with <math>C^{\dagger}C</math> real. Writing <math>C=X+iY</math> with <math>X</math> and <math>Y</math> real symmetric matrices,  <math>C^{\dagger}C=X^2+Y^2+i(XY-YX)</math>. Thus <math>XY=YX</math>. Since <math>X</math> and <math>Y</math> commute, there is a real orthogonal matrix <math>W</math> such that both <math>W X W^{\mathrm T}</math> and <math>W Y W^{\mathrm T}</math> are diagonal. Setting <math>U=W V^{\mathrm T}</math> (a unitary matrix), the matrix <math>UAU^{\mathrm T}</math> is complex diagonal. Pre-multiplying <math>U</math> by a suitable diagonal unitary matrix (which preserves unitarity of <math>U</math>), the diagonal entries of <math>UAU^{\mathrm T}</math> can be made to be real and non-negative as desired. Since their squares are the eigenvalues of <math>A^{\dagger} A</math>, they coincide with the [[singular value]]s of <math>A</math>. (Note, about the eigen-decomposition of a complex symmetric matrix <math>A</math>, the Jordan normal form of <math>A</math> may not be diagonal, therefore <math>A</math> may not be diagonalized by any similarity transformation.)

== Decomposition ==
Using the [[Jordan normal form]], one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices.<ref>{{cite journal | first=A. J.|last= Bosch | title=The factorization of a square matrix into two symmetric matrices | journal=[[American Mathematical Monthly]] | year=1986 | volume=93 | pages=462–464 | doi=10.2307/2323471 | issue=6 | jstor=2323471}}</ref>

Every real [[non-singular matrix]] can be uniquely factored as the product of an [[orthogonal matrix]] and a symmetric [[positive definite matrix]], which is called a [[polar decomposition]]. Singular matrices can also be factored, but not uniquely.

[[Cholesky decomposition]] states that every real positive-definite symmetric matrix <math>A</math> is a product of a lower-triangular matrix <math>L</math> and its transpose,
:<math>A = LL^\textsf{T}</math>. 

If the matrix is symmetric indefinite, it may be still decomposed as <math>PAP^\textsf{T} = LDL^\textsf{T}</math> where <math>P</math> is a permutation matrix (arising from the need to [[pivot element|pivot]]), <math>L</math> a lower unit triangular matrix, and <math>D</math> {{Relevance inline|reason=not referred to in this section|date=December 2015}} is a direct sum of symmetric <math>1 \times 1</math> and <math>2 \times 2</math> blocks, which is called Bunch–Kaufman decomposition <ref>{{cite book | author=G.H. Golub, C.F. van Loan. | title=Matrix Computations | publisher=The Johns Hopkins University Press, Baltimore, London | year=1996}}</ref>

A complex symmetric matrix may not be diagonalizable by similarity; every real symmetric matrix is diagonalizable by a real orthogonal similarity.

Every complex symmetric matrix <math>A</math> can be diagonalized by unitary congruence
:<math>A = Q \Lambda Q^\textsf{T}</math>

where <math>Q</math> is a [[unitary matrix]]. If A is real, the matrix <math>Q</math> is a real [[orthogonal matrix]], (the columns of which are [[eigenvectors]] of <math>A</math>), and <math>\Lambda</math> is real and diagonal (having the [[eigenvalues]] of <math>A</math> on the diagonal). To see orthogonality, suppose <math>x</math> and <math>y</math> are eigenvectors corresponding to distinct eigenvalues <math>\lambda_1</math>, <math>\lambda_2</math>. Then
:<math>\lambda_1 \langle x, y \rangle = \langle Ax, y \rangle = \langle x, Ay \rangle = \lambda_2 \langle x, y \rangle.</math>

Since <math>\lambda_1</math> and <math>\lambda_2</math> are distinct, we have <math>\langle x, y \rangle = 0</math>.

== Hessian ==
Symmetric <math>n \times n</math> matrices of real functions appear as the [[Hessian matrix|Hessians]] of twice continuously differentiable functions of <math>n</math> real variables.

Every [[quadratic form]] <math>q</math> on <math>\mathbb{R}^n</math> can be uniquely written in the form <math>q(\mathbf{x}) = \mathbf{x}^\textsf{T} A \mathbf{x}</math> with a symmetric <math>n \times n</math> matrix <math>A</math>. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of <math>\mathbb{R}^n</math>, "looks like"
:<math>q\left(x_1, \ldots, x_n\right) = \sum_{i=1}^n \lambda_i x_i^2</math>

with real numbers <math>\lambda_i</math>. This considerably simplifies the study of quadratic forms, as well as the study of the level sets <math>\left\{ \mathbf{x} : q(\mathbf{x}) = 1 \right\}</math> which are generalizations of [[conic section]]s.

This is important partly because the second-order behavior of every smooth multi-variable function is described by the quadratic form belonging to the function's Hessian; this is a consequence of [[Taylor's theorem]].

== Symmetrizable matrix ==
An <math>n \times n</math> matrix <math>A</math> is said to be '''symmetrizable''' if there exists an invertible [[diagonal matrix]] <math>D</math> and symmetric matrix <math>S</math> such that <math>A = DS.</math>

The transpose of a symmetrizable matrix is symmetrizable, since <math>A^{\mathrm T}=(DS)^{\mathrm T}=SD=D^{-1}(DSD)</math> and <math>DSD</math> is symmetric. A matrix <math>A=(a_{ij})</math> is symmetrizable if and only if the following conditions are met:
# <math>a_{ij} = 0</math> implies <math>a_{ji} = 0</math> for all <math>1 \le i \le j \le n.</math>
# <math>a_{i_1 i_2} a_{i_2 i_3} \dots a_{i_k i_1} = a_{i_2 i_1} a_{i_3 i_2} \dots a_{i_1 i_k}</math> for any finite sequence <math>\left(i_1, i_2, \dots, i_k\right).</math>

== See also ==
<!--These examples would be more appropriate for square matrix article-->
Other types of [[symmetry]] or pattern in square matrices have special names; see for example:
{{Div col|colwidth=25em}}
* [[Antimetric matrix]]
* [[Centrosymmetric matrix]]
* [[Circulant matrix]]
* [[Covariance matrix]]
* [[Coxeter matrix]]
* [[Hankel matrix]]
* [[Hilbert matrix]]
* [[Persymmetric matrix]]
* [[Skew-symmetric matrix]]
* [[Sylvester's law of inertia]]
* [[Toeplitz matrix]]
{{Div col end}}

See also [[symmetry in mathematics]].

== Notes ==
{{Reflist}}

== References ==
*{{citation|last=Horn|first= Roger A.|last2= Johnson|first2= Charles R.|title= Matrix analysis|edition=2nd| publisher=Cambridge University Press|year= 2013|isbn= 978-0-521-54823-6}}

== External links ==
* {{springer|title=Symmetric matrix|id=p/s091680}}
* [http://farside.ph.utexas.edu/teaching/336k/Newton/node66.html A brief introduction and proof of eigenvalue properties of the real symmetric matrix]
* [https://fylux.github.io/2017/03/07/Symmetric-Triangular-Matrix/ How to implement a Symmetric Matrix in C++]
{{Matrix classes}}
[[Category:Matrices]]