"""Checks if the matrix is a Hermitian matrix."""

import numpy as np

from toqito.matrix_props import is_square


def is_hermitian(mat: np.ndarray, rtol: float = 1e-05, atol: float = 1e-08) -> bool:
    r"""Check if matrix is Hermitian :cite:`WikiHerm`.

    A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose.

    Examples
    ==========

    Consider the following matrix:

    .. math::
        A = \begin{pmatrix}
                2 & 2 +1j & 4 \\
                2 - 1j & 3 & 1j \\
                4 & -1j & 1
            \end{pmatrix}

    our function indicates that this is indeed a Hermitian matrix as it holds that

    .. math::
        A = A^*.

    .. jupyter-execute::

     import numpy as np
     from toqito.matrix_props import is_hermitian

     mat = np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]])

     is_hermitian(mat)

    Alternatively, the following example matrix :math:`B` defined as

    .. math::
        B = \begin{pmatrix}
                1 & 2 & 3 \\
                4 & 5 & 6 \\
                7 & 8 & 9
            \end{pmatrix}

    is not Hermitian.

    .. jupyter-execute::

     import numpy as np
     from toqito.matrix_props import is_hermitian

     mat = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

     is_hermitian(mat)

    References
    ==========
    .. bibliography::
        :filter: docname in docnames



    :param mat: Matrix to check.
    :param rtol: The relative tolerance parameter (default 1e-05).
    :param atol: The absolute tolerance parameter (default 1e-08).
    :return: Return True if matrix is Hermitian, and False otherwise.

    """
    if not is_square(mat):
        return False
    return np.allclose(mat, mat.conj().T, rtol=rtol, atol=atol)
