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Symmetric matrix

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Matrix equal to its transpose

This article is about a matrix symmetric about its diagonal. For a matrix symmetric about its center, seeCentrosymmetric matrix.

For matrices with symmetry over thecomplex number field, seeHermitian matrix.

Inlinear algebra, asymmetric matrix is asquare matrix that is equal to itstranspose. Formally,

$A{\text{ is symmetric}}\iff A=A^{\textsf {T}}.$

Because equal matrices have equal dimensions, only square matrices can be symmetric.

The entries of a symmetric matrix are symmetric with respect to themain diagonal. So if $a_{ij}$ denotes the entry in the $i {\displaystyle i}$ th row and $j {\displaystyle j}$ th column then

$A{\text{ is symmetric}}\iff {\text{ for every }}i,j,\quad a_{ji}=a_{ij}$

for all indices $i {\displaystyle i}$ and $j . {\displaystyle j.}$

Every squarediagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly incharacteristic different from 2, each diagonal element of askew-symmetric matrix must be zero, since each is its own negative.

In linear algebra, areal symmetric matrix represents aself-adjoint operator^[1] represented in anorthonormal basis over areal inner product space. The corresponding object for acomplex inner product space is aHermitian matrix with complex-valued entries, which is equal to itsconjugate 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

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The following $3\times 3$ matrix is symmetric: $A={\begin{bmatrix}1&7&3\\7&4&5\\3&5&2\end{bmatrix}}$ Since $A=A^{\textsf {T}}$ .

Properties

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Basic properties

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

Decomposition into symmetric and skew-symmetric

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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 thedirect sum. Let $X\in {\mbox{Mat}}_{n}$ then $X={\frac {1}{2}}\left(X+X^{\textsf {T}}\right)+{\frac {1}{2}}\left(X-X^{\textsf {T}}\right).$

Notice that ${\textstyle {\frac {1}{2}}\left(X+X^{\textsf {T}}\right)\in {\mbox{Sym}}_{n}}$ and ${\textstyle {\frac {1}{2}}\left(X-X^{\textsf {T}}\right)\in \mathrm {Skew} _{n}}$ . This is true for everysquare matrix $X {\displaystyle X}$ with entries from anyfield whosecharacteristic 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 themain diagonal). Similarly, askew-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

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Any matrixcongruent to a symmetric matrix is again symmetric: if $X {\displaystyle X}$ is a symmetric matrix, then so is $AXA^{\mathrm {T} }$ for any matrix $A {\displaystyle A}$ .

Symmetry implies normality

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A (real-valued) symmetric matrix is necessarily anormal matrix.

Real symmetric matrices

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Denote by $\langle \cdot ,\cdot \rangle$ the standardinner product on $\mathbb {R} ^{n}$ . The real $n\times n$ matrix $A {\displaystyle A}$ is symmetric if and only if $\langle Ax,y\rangle =\langle x,Ay\rangle \quad \forall x,y\in \mathbb {R} ^{n}.$

Since this definition is independent of the choice ofbasis, symmetry is a property that depends only on thelinear operator A and a choice ofinner product. This characterization of symmetry is useful, for example, indifferential geometry, for eachtangent space to amanifold may be endowed with an inner product, giving rise to what is called aRiemannian manifold. Another area where this formulation is used is inHilbert spaces.

The finite-dimensionalspectral theorem says that any symmetric matrix whose entries arereal can bediagonalized by anorthogonal matrix. More explicitly: For every real symmetric matrix $A {\displaystyle A}$ there exists a real orthogonal matrix $Q {\displaystyle Q}$ such that $D=Q^{\mathrm {T} }AQ$ is adiagonal matrix. Every real symmetric matrix is thus,up to choice of anorthonormal basis, a diagonal matrix.

If $A {\displaystyle A}$ and $B {\displaystyle B}$ are $n\times n$ real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix:^[2] there exists a basis of $\mathbb {R} ^{n}$ such that every element of the basis is aneigenvector for both $A {\displaystyle A}$ and $B {\displaystyle B}$ .

Every real symmetric matrix isHermitian, and therefore all itseigenvalues are real. (In fact, the eigenvalues are the entries in the diagonal matrix $D {\displaystyle D}$ (above), and therefore $D {\displaystyle D}$ is uniquely determined by $A {\displaystyle A}$ 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

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A complex symmetric matrix can be 'diagonalized' using aunitary matrix: thus if $A {\displaystyle A}$ is a complex symmetric matrix, there is a unitary matrix $U {\displaystyle U}$ such that $UAU^{\mathrm {T} }$ is a real diagonal matrix with non-negative entries. This result is referred to as theAutonne–Takagi factorization. It was originally proved byLéon Autonne (1915) andTeiji Takagi (1925) and rediscovered with different proofs by several other mathematicians.^[3]^[4] In fact, the matrix $B=A^{\dagger }A$ is Hermitian andpositive semi-definite, so there is a unitary matrix $V {\displaystyle V}$ such that $V^{\dagger }BV$ is diagonal with non-negative real entries. Thus $C=V^{\mathrm {T} }AV$ is complex symmetric with $C^{\dagger }C$ real. Writing $C=X+iY$ with $X {\displaystyle X}$ and $Y {\displaystyle Y}$ real symmetric matrices, $C^{\dagger }C=X^{2}+Y^{2}+i(XY-YX)$ . Thus $XY=YX$ . Since $X {\displaystyle X}$ and $Y {\displaystyle Y}$ commute, there is a real orthogonal matrix $W {\displaystyle W}$ such that both $WXW^{\mathrm {T} }$ and $WYW^{\mathrm {T} }$ are diagonal. Setting $U=WV^{\mathrm {T} }$ (a unitary matrix), the matrix $UAU^{\mathrm {T} }$ is complex diagonal. Pre-multiplying $U {\displaystyle U}$ by a suitable diagonal unitary matrix (which preserves unitarity of $U {\displaystyle U}$ ), the diagonal entries of $UAU^{\mathrm {T} }$ can be made to be real and non-negative as desired. To construct this matrix, we express the diagonal matrix as $UAU^{\mathrm {T} }=\operatorname {diag} (r_{1}e^{i\theta _{1}},r_{2}e^{i\theta _{2}},\dots ,r_{n}e^{i\theta _{n}})$ . The matrix we seek is simply given by $D=\operatorname {diag} (e^{-i\theta _{1}/2},e^{-i\theta _{2}/2},\dots ,e^{-i\theta _{n}/2})$ . Clearly $DUAU^{\mathrm {T} }D=\operatorname {diag} (r_{1},r_{2},\dots ,r_{n})$ as desired, so we make the modification $U'=DU$ . Since their squares are the eigenvalues of $A^{\dagger }A$ , they coincide with thesingular values of $A {\displaystyle A}$ . (Note, about the eigen-decomposition of a complex symmetric matrix $A {\displaystyle A}$ , the Jordan normal form of $A {\displaystyle A}$ may not be diagonal, therefore $A {\displaystyle A}$ may not be diagonalized by any similarity transformation.)

Decomposition

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Using theJordan 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.^[5]

Every realnon-singular matrix can be uniquely factored as the product of anorthogonal matrix and a symmetricpositive definite matrix, which is called apolar decomposition. Singular matrices can also be factored, but not uniquely.

Cholesky decomposition states that every real positive-definite symmetric matrix $A {\displaystyle A}$ is a product of a lower-triangular matrix $L {\displaystyle L}$ 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 $P {\displaystyle P}$ is a permutation matrix (arising from the need topivot), $L {\displaystyle L}$ a lower unit triangular matrix, and $D {\displaystyle D}$ is a direct sum of symmetric $1\times 1$ and $2\times 2$ blocks, which is called Bunch–Kaufman decomposition^[6]

A general (complex) symmetric matrix may bedefective and thus not bediagonalizable. If $A {\displaystyle A}$ is diagonalizable it may be decomposed as $A=Q\Lambda Q^{\textsf {T}}$ where $Q {\displaystyle Q}$ is an orthogonal matrix $QQ^{\textsf {T}}=I$ , and $\Lambda$ is a diagonal matrix of the eigenvalues of $A {\displaystyle A}$ . In the special case that $A {\displaystyle A}$ is real symmetric, then $Q {\displaystyle Q}$ and $\Lambda$ are also real. To see orthogonality, suppose $\mathbf {x}$ and $\mathbf {y}$ are eigenvectors corresponding to distinct eigenvalues $\lambda _{1}$ , $\lambda _{2}$ . Then $\lambda _{1}\langle \mathbf {x} ,\mathbf {y} \rangle =\langle A\mathbf {x} ,\mathbf {y} \rangle =\langle \mathbf {x} ,A\mathbf {y} \rangle =\lambda _{2}\langle \mathbf {x} ,\mathbf {y} \rangle .$

Since $\lambda _{1}$ and $\lambda _{2}$ are distinct, we have $\langle \mathbf {x} ,\mathbf {y} \rangle =0$ .

Hessian

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Symmetric $n\times n$ matrices of real functions appear as theHessians of twice differentiable functions of $n {\displaystyle n}$ real variables (the continuity of the second derivative is not needed, despite common belief to the opposite^[7]).

Everyquadratic form $q {\displaystyle q}$ 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 $A {\displaystyle A}$ . Because of the above spectral theorem, one can then say that every quadratic form, up to the choice of an orthonormal basis of $\mathbb {R} ^{n}$ , "looks like" $q\left(x_{1},\ldots ,x_{n}\right)=\sum _{i=1}^{n}\lambda _{i}x_{i}^{2}$ with 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 ofconic 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 ofTaylor's theorem.

Symmetrizable matrix

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An $n\times n$ matrix $A {\displaystyle A}$ is said to besymmetrizable if there exists an invertiblediagonal matrix $D {\displaystyle D}$ and symmetric matrix $S {\displaystyle S}$ such that $A=DS.$

The transpose of a symmetrizable matrix is symmetrizable, since $A^{\mathrm {T} }=(DS)^{\mathrm {T} }=SD=D^{-1}(DSD)$ and $D S D {\displaystyle DSD}$ is symmetric. A matrix $A=(a_{ij})$ is symmetrizable if and only if the following conditions are met:

$a_{ij}=0$ implies $a_{ji}=0$ for all $1\leq i\leq j\leq n.$
$a_{i_{1}i_{2}}a_{i_{2}i_{3}}\dots a_{i_{k}i_{1}}=a_{i_{2}i_{1}}a_{i_{3}i_{2}}\dots a_{i_{1}i_{k}}$ for any finite sequence $\left(i_{1},i_{2},\dots ,i_{k}\right).$

Notes

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^Jesús Rojo García (1986).Álgebra lineal (in Spanish) (2nd ed.). Editorial AC.ISBN 84-7288-120-2.
^Bellman, Richard (1997).Introduction to Matrix Analysis (2nd ed.). SIAM.ISBN 08-9871-399-4.
^Horn & Johnson 2013, pp. 263, 278
^
See:
- Autonne, L. (1915),"Sur les matrices hypohermitiennes et sur les matrices unitaires",Ann. Univ. Lyon,38:1–77
- Takagi, T. (1925), "On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau",Jpn. J. Math.,1:83–93,doi:10.4099/jjm1924.1.0_83
- Siegel, Carl Ludwig (1943), "Symplectic Geometry",American Journal of Mathematics,65 (1):1–86,doi:10.2307/2371774,JSTOR 2371774, Lemma 1, page 12
- Hua, L.-K. (1944), "On the theory of automorphic functions of a matrix variable I–geometric basis",Amer. J. Math.,66 (3):470–488,doi:10.2307/2371910,JSTOR 2371910
- Schur, I. (1945), "Ein Satz über quadratische Formen mit komplexen Koeffizienten",Amer. J. Math.,67 (4):472–480,doi:10.2307/2371974,JSTOR 2371974
- Benedetti, R.; Cragnolini, P. (1984), "On simultaneous diagonalization of one Hermitian and one symmetric form",Linear Algebra Appl.,57:215–226,doi:10.1016/0024-3795(84)90189-7
^Bosch, A. J. (1986). "The factorization of a square matrix into two symmetric matrices".American Mathematical Monthly.93 (6):462–464.doi:10.2307/2323471.JSTOR 2323471.
^Golub, G.H.;van Loan, C.F. (1996).Matrix Computations. Johns Hopkins University Press.ISBN 0-8018-5413-X.OCLC 34515797.
^Dieudonné, Jean A. (1969). "Theorem (8.12.2)".Foundations of Modern Analysis. Academic Press. p. 180.ISBN 0-12-215550-5.OCLC 576465.

References

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Horn, Roger A.; Johnson, Charles R. (2013),Matrix analysis (2nd ed.), Cambridge University Press,ISBN 978-0-521-54823-6

External links

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