Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if

${\displaystyle A=A^{\mathrm {T} }.}$

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 the entries are written as A = (a_ij), then a_ij = a_ji, for all indices i and j.

The following 3 × 3 matrix is symmetric:

${\begin{bmatrix}1&7&3\\7&4&-5\\3&-5&6\end{bmatrix}}.$

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 operator^[1] 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.

Properties

The sum and difference of two symmetric matrices is again symmetric, but 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. So for integer n, Aⁿ is symmetric if A is symmetric. If A⁻¹ exists, it is symmetric if and only if A is symmetric.

Let Mat_n denote the space of n × n matrices. A symmetric n × n matrix is determined by n(n + 1)/2 scalars (the number of entries on or above the main diagonal). Similarly, a skew-symmetric matrix is determined by n(n − 1)/2 scalars (the number of entries above the main diagonal). If Sym_n denotes the space of n × n symmetric matrices and Skew_n the space of n × n skew-symmetric matrices then Mat_n = Sym_n + Skew_n and Sym_n ∩ Skew_n = {0}, i.e.

${\mbox{Mat}}_{n}={\mbox{Sym}}_{n}\oplus {\mbox{Skew}}_{n},$

where ⊕ denotes the direct sum. Let X ∈ Mat_n then

${\displaystyle X={\frac {1}{2}}(X+X^{\mathrm {T} })+{\frac {1}{2}}(X-X^{\mathrm {T} }).}$

Notice that 1/2(X + X^T) ∈ Sym_n and 1/2(X − X^T) ∈ Skew_n. This is true for every square matrix X with entries from any field whose characteristic is different from 2.

Any matrix congruent to a symmetric matrix is again symmetric: if X is a symmetric matrix then so is AXA^T for any matrix A. A symmetric matrix is necessarily a normal matrix.

Real symmetric matrices

Denote by $\langle \cdot ,\cdot \rangle$ the standard inner product on Rⁿ. The real n-by-n matrix 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 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 symmetric real matrix A there exists a real orthogonal matrix Q such that D = Q^TAQ is a diagonal matrix. Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

If A and B are n × n real symmetric matrices that commute, then they can be simultaneously diagonalized: 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

A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if A is a complex symmetric matrix, there is a unitary matrix $\mathbf {U}$ such that ${\displaystyle \mathbf {U} \mathbf {A} \mathbf {U} ^{\mathrm {T} }}$ is a real diagonal matrix. 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.^[2][3] In fact, the matrix B = A^†A is Hermitian and non-negative, so there is a unitary matrix V such that V^†BV is diagonal with non-negative real entries. Thus C = V^TAV is complex symmetric with C^†C real. Writing C = X + iY with X and Y real symmetric matrices, C^†C = X² + Y² + i(XY − YX). Thus XY = YX. Since X and Y commute, there is a real orthogonal matrix W such that both WXW^T and WYW^T are diagonal. Setting U = WV^T, the matrix UAU^T is complex diagonal. Post-multiplying U by another diagonal matrix the diagonal entries can be made to be real and non-negative. Since their squares are the eigenvalues of A^†A, they coincide with the singular values of A. (Note, about the eigen-decomposition of a complex symmetric matrix A, the Jordan normal form of A may not be diagonal, therefore A 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.^[4]

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, ${\displaystyle A=LL^{\mathrm {T} }}$. If the matrix is symmetric indefinite, it may be still decomposed as ${\displaystyle PAP^{\mathrm {T} }=LDL^{\mathrm {T} }}$ 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 × 1 and 2 × 2 blocks, which is called Bunch-Kaufman decomposition ^[5]

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 A can be diagonalized by unitary congruence

${\displaystyle A=Q\Lambda Q^{\mathrm {T} }}$

where Q is an unitary matrix. If A is real, the matrix Q is a real orthogonal matrix, (the columns of which are eigenvectors of A), and Λ is real and diagonal (having the eigenvalues of A on the diagonal). To see orthogonality, suppose $x$ and $y$ are eigenvectors corresponding to distinct eigenvalues $\lambda _{1}$, $\lambda _{2}$. Then

$\lambda _{1}\langle x,y\rangle =\langle Ax,y\rangle =\langle x,Ay\rangle =\lambda _{2}\langle x,y\rangle$.

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

Hessian

Symmetric n-by-n matrices of real functions appear as the Hessians of twice continuously differentiable functions of n real variables.

Every quadratic form q on Rⁿ can be uniquely written in the form q(x) = x^TAx with a symmetric n-by-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 Rⁿ, "looks like"

$q(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\lambda _{i}x_{i}^{2}$

with real numbers λ_i. This considerably simplifies the study of quadratic forms, as well as the study of the level sets {x : q(x) = 1} 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-by-n matrix A is said to be symmetrizable if there exist an invertible diagonal matrix D and symmetric matrix S such that A = DS. The transpose of a symmetrizable matrix is symmetrizable, since A^T = (DS)^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 $1\leq i\leq j\leq n.$
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 $(i_{1},i_{2},\dots ,i_{k}).$

See also

Other types of symmetry or pattern in square matrices have special names; see for example:

• 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

See also symmetry in mathematics.

Notes

1. Jesús Rojo García (1986). Álgebra lineal (in Spanish) (2^nd. ed.). Editorial AC. ISBN 84 7288 120 2.
2. Horn & Johnson 2013, p. 278
3. 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", Japan. J. Math., 1: 83–93
   • Siegel, Carl Ludwig (1943), "Symplectic Geometry", American Journal of Mathematics, 65: 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: 470–488, doi:10.2307/2371910
   • Schur, I. (1945), "Ein Satz über quadratische formen mit komplexen koeffizienten", Amer. J. Math., 67: 472–480, doi:10.2307/2371974
   • 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
4. 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.
5. G.H. Golub, C.F. van Loan. (1996). Matrix Computations. The Johns Hopkins University Press, Baltimore, London.

References

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

External links

• Hazewinkel, Michiel, ed. (2001), "Symmetric matrix", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• A brief introduction and proof of eigenvalue properties of the real symmetric matrix