Built-in Mathematica Symbol

HermitianMatrixQ

HermitianMatrixQ[m]
tests whether m is a Hermitian matrix.

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

Test if a matrix is explicitly Hermitian:

For a real matrix, this is equivalent to symmetry:

Test if a matrix is explicitly Hermitian:

For a real matrix, this is equivalent to symmetry:

HermitianMatrixQ works with symbolic matrices:

All symbolic quantities are assumed to be complex:

Use a different method for Hermitian matrices:

Construct complex-valued matrices for testing:

For the non-Hermitian complex-valued matrix m, the function just uses Gaussian elimination:

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

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

HermitianMatrixQ[m] is effectively equivalent to m

Hermitian matrices have all real eigenvalues:

This also means that their characteristic polynomials have real coefficients: