matrix_props.is_pseudo_hermitian

Checks if matrix is pseudo hermitian with respect to given signature.

Functions

is_pseudo_hermitian(mat, signature[, rtol, atol])

Check if a matrix is pseudo-Hermitian.

Module Contents

matrix_props.is_pseudo_hermitian.is_pseudo_hermitian(mat, signature, rtol=1e-05, atol=1e-08)

Check if a matrix is pseudo-Hermitian.

A matrix \(H\) is pseudo-Hermitian with respect to a given signature matrix \(\eta\) if it satisfies:

\[\eta H \eta^{-1} = H^{\dagger},\]
where:

  • \(H^{\dagger}\) is the conjugate transpose (Hermitian transpose) of \(H\),

  • \(\eta\) is a Hermitian, invertible matrix.

Examples

Consider the following matrix:

\[\begin{split}H = \begin{pmatrix} 1 & 1+i \\ -1+i & -1 \end{pmatrix}\end{split}\]

with the signature matrix:

\[\begin{split}\eta = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\end{split}\]

Our function confirms that \(H\) is pseudo-Hermitian:

>>> import numpy as np
>>> from toqito.matrix_props import is_pseudo_hermitian
>>> H = np.array([[1, 1+1j], [-1+1j, -1]])
>>> eta = np.array([[1, 0], [0, -1]])
>>> is_pseudo_hermitian(H, eta)
True

However, the following matrix \(A\)

\[\begin{split}A = \begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}\end{split}\]

is not pseudo-Hermitian with respect to the same signature matrix:

>>> A = np.array([[1, 1j], [-1j, 1]])
>>> is_pseudo_hermitian(A, eta)
False

References

Parameters:

  • mat (numpy.ndarray) – The matrix to check.

  • signature (numpy.ndarray) – The signature matrix \(\eta\), which must be Hermitian and invertible.

  • rtol (float) – The relative tolerance parameter (default 1e-05).

  • atol (float) – The absolute tolerance parameter (default 1e-08).

Raises:

ValueError – If signature is not Hermitian or not invertible.

Returns:

Return True if the matrix is pseudo-Hermitian, and False otherwise.

Return type:

bool