Editing Symmetric matrix

{{Short description|Matrix equal to its transpose
}}
{{about|a matrix symmetric about its diagonal|a matrix symmetric about its center| Centrosymmetric matrix}}
{{For|matrices with symmetry over the [[complex number]] field|Hermitian matrix}}
{{Use American English|date=January 2019}}
[[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= es|edition=2nd|publisher=Editorial AC|year=1986|isbn=84-7288-120-2}}</ref> represented in an [[orthonormal basis]] 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 display="block">A =
  \begin{bmatrix}
    1 & 7 & 3 \\
    7 & 4 & 5 \\
    3 & 5 & 2
  \end{bmatrix}</math>
Since <math>A=A^\textsf{T}</math>.

== Properties ==
===Basic properties===
* The sum and difference of two symmetric matrices is 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 any 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.
* Rank of a symmetric matrix <math>A</math> is equal to the number of non-zero eigenvalues of <math>A</math>.

===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 display="block">\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 display="block">X = \frac{1}{2}\left(X + X^\textsf{T}\right) + \frac{1}{2}\left(X - X^\textsf{T}\right).</math>

Notice that <math display="inline">\frac{1}{2}\left(X + X^\textsf{T}\right) \in \mbox{Sym}_n</math> and <math display="inline">\frac{1}{2} \left(X - X^\textsf{T}\right) \in \mathrm{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]].

=== 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 display="block">\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 real symmetric 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 real 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 by an orthogonal matrix:<ref>{{Cite book|first=Richard |last=Bellman|title=Introduction to Matrix Analysis |language= en|edition=2nd|publisher=SIAM|year=1997|isbn=08-9871-399-4}}</ref> 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 {{anchor|Complex}}===
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>{{harvnb|Horn|Johnson|2013|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|url=https://gallica.bnf.fr/ark:/12148/bpt6k69553b}}
*{{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= Jpn. J. Math.|volume= 1 |year=1925|pages= 83–93|doi= 10.4099/jjm1924.1.0_83|doi-access= free}}
*{{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|doi-access=free}}
</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. To construct this matrix, we express the diagonal matrix as <math>UAU^\mathrm T = \operatorname{diag}(r_1 e^{i\theta_1},r_2 e^{i\theta_2}, \dots, r_n e^{i\theta_n})</math>. The matrix we seek is simply given by <math>D = \operatorname{diag}(e^{-i\theta_1/2},e^{-i\theta_2/2}, \dots, e^{-i\theta_n/2})</math>. Clearly <math>DUAU^\mathrm TD = \operatorname{diag}(r_1, r_2, \dots, r_n)</math> as desired, so we make the modification <math>U' = DU</math>. 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 display="block">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> 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-link1=Gene H. Golub |last1=Golub |first1=G.H. |author2-link=Charles F. Van Loan |last2=van Loan |first2=C.F. | title=Matrix Computations | publisher=Johns Hopkins University Press| year=1996 |isbn=0-8018-5413-X |oclc=34515797 }}</ref>

A general (complex) symmetric matrix may be [[defective matrix|defective]] and thus not be [[diagonalizable]]. If <math>A</math> is diagonalizable it may be decomposed as
<math display="block">A = Q \Lambda Q^\textsf{T}</math>
where <math>Q</math> is an orthogonal matrix <math>Q Q^\textsf{T} = I</math>, and <math>\Lambda</math> is a diagonal matrix of the eigenvalues of <math>A</math>. In the special case that <math>A</math> is real symmetric, then <math>Q</math> and <math>\Lambda</math> are also real. To see orthogonality, suppose <math>\mathbf x</math> and <math>\mathbf y</math> are eigenvectors corresponding to distinct eigenvalues <math>\lambda_1</math>, <math>\lambda_2</math>. Then
<math display="block">\lambda_1 \langle \mathbf x, \mathbf y \rangle = \langle A \mathbf x, \mathbf y \rangle = \langle \mathbf x, A \mathbf y \rangle = \lambda_2 \langle \mathbf x, \mathbf y \rangle.</math>

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

== Hessian ==
Symmetric <math>n \times n</math> matrices of real functions appear as the [[Hessian matrix|Hessians]] of twice differentiable functions of <math>n</math> real variables (the continuity of the second derivative is not needed, despite common belief to the opposite<ref>{{Cite book |last=Dieudonné |first=Jean A. |title=Foundations of Modern Analysis |publisher=Academic Press |year=1969 |chapter=Theorem (8.12.2) |page=180 |isbn=0-12-215550-5 |oclc=576465}}</ref>). 

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>\R^n</math>, "looks like"
<math display="block">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}}
* [[Skew-symmetric matrix]] (also called ''antisymmetric'' or ''antimetric'') 
* [[Centrosymmetric matrix]]
* [[Circulant matrix]]
* [[Covariance matrix]]
* [[Coxeter matrix]]
* [[GCD matrix]]
* [[Hankel matrix]]
* [[Hilbert matrix]]
* [[Persymmetric matrix]]
* [[Sylvester's law of inertia]]
* [[Toeplitz matrix]]
* [[Transpositions matrix]]
{{Div col end}}

See also [[symmetry in mathematics]].

== Notes ==
{{Reflist}}

== References ==
{{refbegin}}
*{{citation|last1=Horn|first1= Roger A.|last2= Johnson|first2= Charles R.|title= Matrix analysis|edition=2nd| publisher=Cambridge University Press|year= 2013|isbn= 978-0-521-54823-6}}
{{refend}}

== 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}}
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[[Category:Matrices (mathematics)]]