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{{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|indent =:|equation = A text{ is symmetric} iff A = A^textsf{T}.|cellpadding= 6|border colour = #0073CF|background colour = #F5FFFA}}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 a_{ij} denotes the entry in the ith row and jth column then{{Equation box 1|indent = :|equation = A text{ is symmetric} iff text{ for every }i,j,quad a_{ji} = a_{ij}|cellpadding= 6|border colour = #0073CF|background colour = #F5FFFA}}for all indices i and j.Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in 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 symmetric matrix represents a self-adjoint operatorBOOK, Jesús Rojo García, Álgebra lineal, es, 2nd, Editorial AC, 1986, 84-7288-120-2, represented in an orthonormal basis over a real inner product space. The corresponding object for a 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 3 times 3 matrix is symmetric:A = Since A=A^textsf{T}.

Properties

Basic properties

• The sum and difference of two symmetric matrices is symmetric.
• This is not always true for the product: given symmetric matrices A and B, then AB is symmetric if and only if A and B commute, i.e., if AB=BA.
• For any integer n, A^n is symmetric if A is symmetric.
• If A^{-1} exists, it is symmetric if and only if A is symmetric.
• Rank of a symmetric matrix A is equal to the number of non-zero eigenvalues of A.

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 mbox{Mat}_n denote the space of n times n matrices. If mbox{Sym}_n denotes the space of n times n symmetric matrices and mbox{Skew}_n the space of n times n skew-symmetric matrices then mbox{Mat}_n = mbox{Sym}_n + mbox{Skew}_n and mbox{Sym}_n cap mbox{Skew}_n = {0}, i.e.mbox{Mat}_n = mbox{Sym}_n oplus mbox{Skew}_n , where oplus denotes the direct sum. Let X in mbox{Mat}_n thenX = frac{1}{2}left(X + X^textsf{T}right) + frac{1}{2}left(X - X^textsf{T}right).Notice that frac{1}{2}left(X + X^textsf{T}right) in mbox{Sym}_n and frac{1}{2} left(X - X^textsf{T}right) in mathrm{Skew}_n. This is true for every square matrix X with entries from any field whose characteristic is different from 2.A symmetric n times n matrix is determined by tfrac{1}{2}n(n+1) scalars (the number of entries on or above the main diagonal). Similarly, a skew-symmetric matrix is determined by tfrac{1}{2}n(n-1) scalars (the number of entries above the main diagonal).

Matrix congruent to a symmetric matrix

Any matrix congruent to a symmetric matrix is again symmetric: if X is a symmetric matrix, then so is A X A^{mathrm T} for any matrix A.

Symmetry implies normality

A (real-valued) symmetric matrix is necessarily a normal matrix.

Real symmetric matrices

Denote by langle cdot,cdot rangle the standard inner product on mathbb{R}^n. The real n times n matrix A is symmetric if and only iflangle Ax, y rangle = langle x, Ay rangle quad forall x, y in mathbb{R}^n.Since this definition is independent of the choice of 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 spaces.The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: For every real symmetric matrix A there exists a real orthogonal matrix Q such that D = Q^{mathrm T} A Q is a diagonal matrix. Every real symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.If A and B are n times n real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix:BOOK, Richard Bellman, Introduction to Matrix Analysis, en, 2nd, SIAM, 1997, 08-9871-399-4, there exists a basis of mathbb{R}^n such that every element of the basis is an eigenvector for both A and B.Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the diagonal matrix D (above), and therefore D is uniquely determined by A 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 A is a complex symmetric matrix, there is a unitary matrix U such that U A U^{mathrm T} 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.BOOK, R.A., Horn, C.R., Johnson, Matrix analysis, 2013, 2nd, Cambridge University Press, 2978290, pp. 263, 278, 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}}

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.JOURNAL, A. J., Bosch, The factorization of a square matrix into two symmetric matrices, American Mathematical Monthly, 1986, 93, 462–464, 10.2307/2323471, 6, 2323471, 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 A is a product of a lower-triangular matrix L and its transpose,A = LL^textsf{T}.If the matrix is symmetric indefinite, it may be still decomposed as PAP^textsf{T} = LDL^textsf{T} where P is a permutation matrix (arising from the need to pivot), L a lower unit triangular matrix, and D is a direct sum of symmetric 1 times 1 and 2 times 2 blocks, which is called Bunch–Kaufman decomposition BOOK, G.H. Golub, C.F. van Loan., Matrix Computations, The Johns Hopkins University Press, Baltimore, London, 1996, A general (complex) symmetric matrix may be defective and thus not be diagonalizable. If A is diagonalizable it may be decomposed asA = Q Lambda Q^textsf{T}where Q is an orthogonal matrix Q Q^textsf{T} = I, and Lambda is a diagonal matrix of the eigenvalues of A. In the special case that A is real symmetric, then Q and Lambda are also real. To see orthogonality, suppose mathbf x and mathbf y are eigenvectors corresponding to distinct eigenvalues lambda_1, lambda_2. Thenlambda_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.Since lambda_1 and lambda_2 are distinct, we have langle mathbf x, mathbf y rangle = 0.

Hessian

Symmetric n times n matrices of real functions appear as the Hessians of twice differentiable functions of n real variables (the continuity of the second derivative is not needed, despite common belief to the oppositeBOOK, Dieudonné, Jean A., Foundations of Modern Analysis, Academic Press, 1969, 978-1443724265, Enlarged and Corrected printing, Theorem (8.12.2), p. 180, en, ). Every quadratic form q on mathbb{R}^n can be uniquely written in the form q(mathbf{x}) = mathbf{x}^textsf{T} A mathbf{x} with a symmetric n times n matrix A. Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of R^n, "looks like"qleft(x_1, ldots, x_nright) = sum_{i=1}^n lambda_i x_i^2with real numbers lambda_i. This considerably simplifies the study of quadratic forms, as well as the study of the level sets left{ mathbf{x} : q(mathbf{x}) = 1 right} which are generalizations of conic sections.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 n times n matrix A is said to be symmetrizable if there exists an invertible diagonal matrix D and symmetric matrix S such that A = DS.The transpose of a symmetrizable matrix is symmetrizable, since A^{mathrm T}=(DS)^{mathrm T}=SD=D^{-1}(DSD) and DSD is symmetric. A matrix A=(a_{ij}) is symmetrizable if and only if the following conditions are met:

1. a_{ij} = 0 implies a_{ji} = 0 for all 1 le i le j le n.
2. 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} for any finite sequence left(i_1, i_2, dots, i_kright).

See also

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

• {{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}}
• A brief introduction and proof of eigenvalue properties of the real symmetric matrix
• How to implement a Symmetric Matrix in C++

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