Built-in Mathematica Symbol

SymmetricMatrixQ

SymmetricMatrixQ[m]
tests whether m is a symmetric matrix.

SymmetricMatrixQ[m] gives True if m is explicitly symmetric, and gives False if it is a matrix that is not symmetric.

Test if a matrix is explicitly symmetric:

SymmetricMatrixQ works with symbolic matrices:

Use a different method for symmetric matrices:

Construct real-valued matrices for testing:

For the non-symmetric matrix m, the function just uses Gaussian elimination:

For the symmetric indefinite matrix mi, the function tries the Cholesky method first:

For the symmetric positive definite matrix mp, the function succeeds with the Cholesky method:

Determine if a sparse matrix is structurally symmetric:

For real-valued matrices, a matrix is Hermitian if and only if it is symmetric:

SymmetricMatrixQ[m] is effectively equivalent to m

Real-valued symmetric matrices have all real eigenvalues:

This also means that their characteristic polynomials have real coefficients:

A complex symmetric matrix is not Hermitian: