Symmetric matrix

In linear algebra, a symmetric matrix is a matrix, A, that is equal to its transpose:

$A=A^{\textrm {T}},\,\!$

This implies that A is a square matrix. The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). 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:

${\begin{bmatrix}1&2&3\\2&4&-5\\3&-5&6\end{bmatrix}}$

A matrix is called skew-symmetric if its transpose is the same as its negative. Thus the following 3×3 matrix is skew-symmetric:

${\begin{bmatrix}0&-3&4\\3&0&-5\\-4&5&0\end{bmatrix}}$

Every diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. The following matrix is neither symmetric nor skew-symmetric:

${\begin{bmatrix}1&-4&2\\5&1&-4\\-3&5&1\end{bmatrix}}$

In pure linear algebra, a symmetric matrix represents a self-adjoint operator 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, it is generally assumed that a symmetric matrix has real-valued entries.

Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodation for them.

Properties

One of the basic theorems concerning such matrices is the finite-dimensional spectral theorem, which says that any symmetric matrix whose entries are real can be diagonalized by an orthogonal matrix. More explicitly: to 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.

Another way of stating the real spectral theorem is that the eigenvectors of a symmetric matrix are orthogonal. More precisely, a matrix is symmetric if and only if it has an orthonormal basis of eigenvectors.

Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real. (In fact, the eigenvalues are the entries in the above diagonal matrix D, 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.

Every square real matrix X can be written in a unique way as the sum of a symmetric and a skew-symmetric matrix. This is done in the following way:

$X={\frac {1}{2}}\left(X+X^{\textrm {T}}\right)+{\frac {1}{2}}\left(X-X^{\textrm {T}}\right).$

(This is true more generally for every square matrix X with entries from any field whose characteristic is different from 2.)

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. Two real symmetric matrices commute if and only if they have the same eigenspaces.

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.

Denote with 〈,〉 the standard inner product on Rⁿ. The real n-by-n matrix A is symmetric if and only if

$\langle Ax,y\rangle =\langle x,Ay\rangle \quad {\mbox{for all }}x,y\in {\mathbb {R}}^{n}$.

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. (Bosch, 1986)

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 be also factored, but not uniquely.

A symmetric $n\times n$ matrix is determined by ${\tfrac {n(n+1)}{2}}$ scalars. Similarly, an anti-symmetric matrix is determined by ${\tfrac {n(n-1)}{2}}$ scalars.

Occurrence

Symmetric real n-by-n matrices appear as the Hessian 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.

See also

• Hermitian matrix
• Normal matrix
• Hilbert matrix
• Self-adjoint operator

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

• Circulant matrix
• Hankel matrix
• Toeplitz matrix
• Centrosymmetric matrix
• Null symmetric matrix

See also symmetry in mathematics.

References

• A. J. Bosch (1986). "The factorization of a square matrix into two symmetric matrices". American Mathematical Monthly. 93: 462–464.