Symmetric Matrix -- From Wolfram MathWorld

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A symmetric matrix is a square matrix that satisfies

A^(T)=A, (1)

where A^(T) denotes the transpose, so a_(ij)=a_(ji). This also implies

A^(-1)A^(T)=I, (2)

where I is the identity matrix. For example,

A=[4 1; 1 -2] (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 m can be tested to see if it is symmetric in the Wolfram Language using SymmetricMatrixQ[m].

Written explicitly, the elements of a symmetric matrix A have the form

[a_(11) a_(12) ... a_(1n); a_(12) a_(22) ... a_(2n); | | ... |; a_(1n) a_(2n) ... a_(nn)]. (4)

The symmetric part of any matrix may be obtained from

A_S=1/2(A+A^(T)). (5)

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

A=QDQ^(T), (6)

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

AQ=QD, (7)

which is equivalent to

AQ_n=lambda_nQ_n (8)

for all n, where lambda_n=D_(nn). Therefore, the diagonal elements of D are the eigenvalues of A, and the columns of Q are the corresponding eigenvectors.

The numbers of symmetric matrices of order n on s symbols are s, s^3, s^6, s^(10), ..., s^(n(n-1)/2). Therefore, for (0,1)-matrices, the numbers of distinct symmetric matrices of orders n=1, 2, ... are 2, 8, 64, 1024, ... (OEIS A006125).

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