"""Is the matrix idempotent."""
import numpy as np
from toqito.matrix_props import is_square
[docs]
def is_idempotent(mat: np.ndarray, rtol: float = 1e-05, atol: float = 1e-8) -> bool:
r"""
Check if matrix is the idempotent matrix [WikIdempotent]_.
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}
>>> from toqito.matrix_props import is_idempotent
>>> import numpy as np
>>> 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.
>>> from toqito.matrix_props import is_idempotent
>>> import numpy as np
>>> mat = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> is_idempotent(mat)
False
References
==========
.. [WikIdempotent] Wikipedia: Idempotent matrix
https://en.wikipedia.org/wiki/Idempotent_matrix
: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)