Construct a Hermitian matrix:

Show the elements:

Normal can convert a HermitianMatrix to its ordinary representation:

Construct a Hermitian matrix from its upper-triangular entries:

Show the elements:

The matrix can also be constructed from its lower-triangular entries:

The Hilbert matrix is a Hermitian matrix:

Complex reflection matrices are both Hermitian and unitary:

HermitianMatrix objects include properties that give information about the matrix:

The "Summary" property gives a brief summary of information about the matrix:

The "StructuredAlgorithms" property lists the functions that have structured algorithms:

Matrices drawn from GaussianUnitaryMatrixDistribution are Hermitian:

Matrices drawn from GaussianSymplecticMatrixDistribution are Hermitian:

The Pauli matrices are Hermitian matrices:

A Hermitian positive-definite matrix defines an inner product by :

Verify that is positive definite:

Orthogonalize the standard basis of to find an orthonormal basis:

Confirm that this basis is orthonormal with respect to the inner product :

The conjugate transpose of a Hermitian matrix equals the original matrix:

A real symmetric matrix is also a Hermitian matrix:

A Hermitian matrix can be represented using SymmetrizedArray or HermitianMatrix:

The two representations are equal, but support different algorithms:

SymmetrizedArray supports tensorial operations such as D, Flatten, Inner, and Outer:

HermitianMatrix supports matrix-specific operations such as KroneckerProduct:

A real Hermitian matrix can also be represented using SymmetricMatrix:

This is not true of a complex Hermitian matrix: