toqito.matrix_props.is_pseudo_hermitian
=======================================

.. py:module:: toqito.matrix_props.is_pseudo_hermitian

.. autoapi-nested-parse::

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





Module Contents
---------------

.. py:function:: 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.

   .. rubric:: Examples

   Consider the following matrix:

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

   with the signature matrix:

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

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

   ```python exec="1" source="above"
   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]])

   print(is_pseudo_hermitian(H, eta))
   ```

   However, the following matrix \(A\)

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

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

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_props import is_pseudo_hermitian
   eta = np.array([[1, 0], [0, -1]])
   A = np.array([[1, 1j], [-1j, 1]])

   print(is_pseudo_hermitian(A, eta))
   ```

   :raises ValueError: If `signature` is not Hermitian or not invertible.

   :param mat: The matrix to check.
   :param signature: The signature matrix \(\eta\), which must be Hermitian and invertible.
   :param rtol: The relative tolerance parameter (default 1e-05).
   :param atol: The absolute tolerance parameter (default 1e-08).

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