Symmetric matrix explained

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

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 i th row and j th column then

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 operator^[1] 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 x 3 matrix is symmetric:Since A = A ^{sf{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

Mat _{n} denote the space of n x n matrices. If Sym _{n} denotes the space of n x n symmetric matrices and Skew _{n} the space of n x n skew-symmetric matrices then Mat _{n} = Sym _{n} + Skew _{n} and Sym _{n} \cap Skew _{n} = \{ 0 \} , i.e.$$\backslash mbox\_n\; =\; \backslash mbox\_n\; \backslash oplus\; \backslash mbox\_n,$$where ⊕ denotes the direct sum. Let X \in Mat _{n} then$$X\; =\; \backslash frac\backslash left(X\; +\; X^\backslash textsf\backslash right)\; +\; \backslash frac\backslash left(X\; -\; X^\backslash textsf\backslash right).$$X with entries from any field whose characteristic is different from 2.

A symmetric

n x 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 ^{T} for any matrix A .

Symmetry implies normality

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

Real symmetric matrices

Denote by

\langle ⋅ , ⋅ \rangle the standard inner product on R ^{n} . The real n x n matrix A is symmetric if and only if$$\backslash langle\; Ax,\; y\; \backslash rangle\; =\; \backslash langle\; x,\; Ay\; \backslash rangle\; \backslash quad\; \backslash forall\; x,\; y\; \backslash in\; \backslash mathbb^n.$$

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 ^{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 x n real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix:^[2] there exists a basis of 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 U such that U A U ^{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.^{[3] [4]} In fact, the matrix B = A ^{\dagger} A is Hermitian and positive semi-definite, so there is a unitary matrix V such that V ^{\dagger} B V is diagonal with non-negative real entries. Thus C = V ^{T} A V is complex symmetric with C ^{\dagger} C real. Writing C = X + iY with X and Y real symmetric matrices, C ^{\dagger} C = X ^{2+Y} ^{2+i(XY-YX)} . Thus XY = YX . Since X and Y commute, there is a real orthogonal matrix W such that both W X W ^{T} and W Y W ^{T} are diagonal. Setting U = W V ^{T} (a unitary matrix), the matrix UAU ^{T} is complex diagonal. Pre-multiplying U by a suitable diagonal unitary matrix (which preserves unitarity of U ), the diagonal entries of UAU ^{T} can be made to be real and non-negative as desired. To construct this matrix, we express the diagonal matrix as UAU ^{T=} \operatorname{diag}(r _{1} e ^{i\theta _{1}} ,r _{2} e ^{i\theta _{2}} , ... , 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}} , ... , e ^{- i\theta _{n/2}} ) . Clearly DUAU ^{TD=} \operatorname{diag}(r _{1,} r _{2,} ... , r _{n)} as desired, so we make the modification U' = DU . Since their squares are the eigenvalues of A ^{\dagger} 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.^[5]

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,$$A\; =\; LL^\backslash textsf.$$

If the matrix is symmetric indefinite, it may be still decomposed as

PAP ^{sf{T}=} LDL ^{sf{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 x 1 and 2 x 2 blocks, which is called Bunch–Kaufman decomposition ^[6]

A general (complex) symmetric matrix may be defective and thus not be diagonalizable. If

A is diagonalizable it may be decomposed as$$A\; =\; Q\; \backslash Lambda\; Q^\backslash textsf$$where Q is an orthogonal matrix Q Q ^{sf{T}=} I , and Λ is a diagonal matrix of the eigenvalues of A . In the special case that A is real symmetric, then Q and Λ are also real. To see orthogonality, suppose x and y are eigenvectors corresponding to distinct eigenvalues λ _{1} , λ _{2} . Then$$\backslash lambda\_1\; \backslash langle\; \backslash mathbf\; x,\; \backslash mathbf\; y\; \backslash rangle\; =\; \backslash langle\; A\; \backslash mathbf\; x,\; \backslash mathbf\; y\; \backslash rangle\; =\; \backslash langle\; \backslash mathbf\; x,\; A\; \backslash mathbf\; y\; \backslash rangle\; =\; \backslash lambda\_2\; \backslash langle\; \backslash mathbf\; x,\; \backslash mathbf\; y\; \backslash rangle.$$

Since

λ _{1} and λ _{2} are distinct, we have \langle x, y \rangle = 0 .

Hessian

Symmetric

n x 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 opposite^[7]). q on R ^{n} can be uniquely written in the form q( x ) = x ^{sf{T}A} x with a symmetric n x 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"$$q\backslash left(x\_1,\; \backslash ldots,\; x\_n\backslash right)\; =\; \backslash sum\_^n\; \backslash lambda\_i\; x\_i^2$$with real numbers λ _{i} . This considerably simplifies the study of quadratic forms, as well as the study of the level sets \left\{ x : q( 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 x 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 ^{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:a _{ij} = 0 implies a _{ji} = 0 for all 1 \le i \le j \le n. a _{i _{1} i _{2}} a _{i _{2} i _{3}} ... a _{i _{k} i _{1}} = a _{i _{2} i _{1}} a _{i _{3} i _{2}} ... a _{i _{1} i _{k}} for any finite sequence \left(i _{1,} i _{2,} ... , i _{k\right).}

See also

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

• 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

See also symmetry in mathematics.

External links

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

Notes and References

1. Book: Jesús Rojo García. Álgebra lineal . es. 2nd. Editorial AC. 1986. 84-7288-120-2.
2. Book: Richard Bellman. Introduction to Matrix Analysis . en. 2nd. SIAM. 1997. 08-9871-399-4.
3. Book: R.A.. Horn. C.R.. Johnson. Matrix analysis . 2013 . 2nd . Cambridge University Press . 2978290. pp. 263, 278.
4. See:
5. A. J.. Bosch . The factorization of a square matrix into two symmetric matrices . . 1986 . 93 . 462–464 . 10.2307/2323471 . 6 . 2323471.
6. Book: G.H. Golub, C.F. van Loan. . Matrix Computations . The Johns Hopkins University Press, Baltimore, London . 1996.
7. Book: Dieudonné, Jean A. . Foundations of Modern Analysis . Academic Press . 1969 . 978-1443724265 . Enlarged and Corrected printing . Theorem (8.12.2), p. 180 . en.