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

.. py:module:: matrix_props.is_pseudo_unitary

.. autoapi-nested-parse::

   Checks if matrix is pseudo unitary.



Functions
---------

.. autoapisummary::

   matrix_props.is_pseudo_unitary.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

   .. math::
       A^* J A = J,

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

   .. rubric:: Examples

   Consider the following matrix:

   .. math::
       A = \begin{pmatrix}
           cosh(1) & sinh(1) \\
           sinh(1) & cosh(1)
       \end{pmatrix}

   with the signature matrix:

   .. math::
       J = \begin{pmatrix}
           1 & 0 \\
           0 & -1
       \end{pmatrix}

   Our function confirms that :math:`A` is pseudo-unitary.

   .. jupyter-execute::

    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)]])

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

   However, the following matrix :math:B

   .. math::
       B = \begin{pmatrix}
           1 & 0 \\
           1 & 1
       \end{pmatrix}

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

   .. jupyter-execute::

    import numpy as np
    from toqito.matrix_props import is_pseudo_unitary

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

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

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :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).
   :raises ValueError: When p < 0 or q < 0.
   :return: Return :code:True if the matrix is pseudo-unitary, and :code:False otherwise.