Symmetric Matrix

A symmetric matrix is a square matrix that satisfies

(1)

where denotes the transpose, so . This also implies

(2)

where is the identity matrix. For example,

(3)

is a symmetric matrix. Hermitian matrices are a useful generalization of symmetric matrices for complex matrices.

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 elements of a symmetric matrix have the form

(4)

The symmetric part of any matrix may be obtained from

(5)

A matrix is symmetric if it can be expressed in the form

(6)

where is an orthogonal matrix and is a diagonal matrix. This is equivalent to the matrix equation

(7)

which is equivalent to

(8)

for all , where . Therefore, the diagonal elements of are the eigenvalues of , and the columns of are the corresponding eigenvectors.

The numbers of symmetric matrices of order on symbols are , , , , ..., . Therefore, for (0,1)-matrices, the numbers of distinct symmetric matrices of orders , 2, ... are 2, 8, 64, 1024, ... (OEIS A006125).

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