toqito.matrix_props.is_pseudo_unitary
=====================================

.. py:module:: toqito.matrix_props.is_pseudo_unitary

.. autoapi-nested-parse::

   Checks if matrix is pseudo unitary.





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

.. py:function:: is_pseudo_unitary(mat, p, q, rtol = 1e-05, atol = 1e-08)

   Check if a matrix is pseudo-unitary.

   A matrix A of size (p+q)x(p+q) is pseudo-unitary with respect to a given signature matrix J if it satisfies

   \[
       A^* J A = J,
   \]

   where:

   - \(A^*\) is the conjugate transpose (Hermitian transpose) of \(A\),
   - \(J\) is a diagonal matrix with first \(p\) diagonal entries equal to 1 and next \(q\)
       diagonal entries equal to -1

   .. rubric:: Examples

   Consider the following matrix:

   \[
       A = \begin{pmatrix}
           cosh(1) & sinh(1) \\
           sinh(1) & cosh(1)
       \end{pmatrix}
   \]

   with the signature matrix:

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

   Our function confirms that \(A\) is pseudo-unitary.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_props import is_pseudo_unitary

   A = np.array([[np.cosh(1), np.sinh(1)], [np.sinh(1), np.cosh(1)]])

   print(is_pseudo_unitary(A, p=1, q=1))
   ```


   However, the following matrix \(B\)

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

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

   ```python exec="1" source="above"
   import numpy as np
   from toqito.matrix_props import is_pseudo_unitary

   B = np.array([[1, 0], [1, 1]])

   print(is_pseudo_unitary(B, p=1, q=1))
   ```

   :raises ValueError: When p < 0 or q < 0.

   :param mat: The matrix to check.
   :param p: Number of positive entries in the signature matrix.
   :param q: Number of negative entries in the signature matrix.
   :param rtol: The relative tolerance parameter (default 1e-05).
   :param atol: The absolute tolerance parameter (default 1e-08).

   :returns: code:True if the matrix is pseudo-unitary, and :code:False otherwise.
   :rtype: Return