"""Checks if the matrix is an idempotent matrix."""

import numpy as np

from toqito.matrix_props import is_square


def is_idempotent(mat: np.ndarray, rtol: float = 1e-05, atol: float = 1e-8) -> bool:
    r"""Check if matrix is the idempotent matrix :cite:`WikiIdemPot`.

    An *idempotent matrix* is a square matrix, which, when multiplied by itself, yields itself.
    That is, the matrix :math:`A` is idempotent if and only if :math:`A^2 = A`.

    Examples
    ==========

    The following is an example of a :math:`2 x 2` idempotent matrix:

    .. math::
        A = \begin{pmatrix}
            3 & -6 \\
            1 & -2
        \end{pmatrix}

    .. jupyter-execute::

     import numpy as np
     from toqito.matrix_props import is_idempotent

     mat = np.array([[3, -6], [1, -2]])

     is_idempotent(mat)


    Alternatively, the following matrix

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

    is not idempotent.

    .. jupyter-execute::

     import numpy as np
     from toqito.matrix_props import is_idempotent

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

     is_idempotent(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 :code:`True` if matrix is the idempotent matrix, and
            :code:`False` otherwise.

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