Symmetric Matrix

A symmetric matrix is a square matrix that satisfies

(1)

where denotes the transpose, so . If is a nonsingular matrix, this also implies

(2)

where is the identity matrix. For example,

(3)

is a symmetric matrix. For a real matrix, symmetry is equivalent to being a Hermitian matrix. For complex matrices, these are different conditions.

A matrix that is not symmetric is said to be an asymmetric matrix, not to be confused with an antisymmetric matrix.

A matrix can be tested to see if it is symmetric in the Wolfram Language using SymmetricMatrixQ[m].

Written explicitly, the entries of a symmetric matrix have the form

(4)

The symmetric part of any real matrix or complex matrix that is a square matrix is given by

(5)

A real matrix is symmetric iff it can be expressed in the form

(6)

where is an orthogonal matrix and is a real matrix that is a diagonal matrix. The spectral theorem for a real symmetric matrix guarantees this decomposition, which is equivalent to the matrix equation

(7)

which is equivalent to

(8)

for , 2, ..., , where is , is the th column of , and . Therefore, the diagonal entries of are the eigenvalues of , and the columns of are the corresponding eigenvectors.

An symmetric matrix with entries chosen from distinct symbols is determined by its entries on and above the diagonal. Thus there are such matrices, where and are positive integers. Therefore, for (0,1)-matrices, the numbers of distinct symmetric matrices of orders , 2, ... are 2, 8, 64, 1024, ... (OEIS A006125).