"""Is matrix the identity matrix."""
import numpy as np
from toqito.matrix_props import is_square
[docs]
def is_identity(mat: np.ndarray, rtol: float = 1e-05, atol: float = 1e-8) -> bool:
r"""
Check if matrix is the identity matrix [WikIdentity]_.
For dimension :math:`n`, the :math:`n \times n` identity matrix is defined as
.. math::
I_n =
\begin{pmatrix}
1 & 0 & 0 & \ldots & 0 \\
0 & 1 & 0 & \ldots & 0 \\
0 & 0 & 1 & \ldots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \ldots & 1
\end{pmatrix}.
Examples
==========
Consider the following matrix:
.. math::
A = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
our function indicates that this is indeed the identity matrix of dimension
3.
>>> from toqito.matrix_props import is_identity
>>> import numpy as np
>>> mat = np.eye(3)
>>> is_identity(mat)
True
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 an identity matrix.
>>> from toqito.matrix_props import is_identity
>>> import numpy as np
>>> mat = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> is_identity(mat)
False
References
==========
.. [WikIdentity] Wikipedia: Identity matrix
https://en.wikipedia.org/wiki/Identity_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 identity matrix, and
:code:`False` otherwise.
"""
if not is_square(mat):
return False
id_mat = np.eye(len(mat))
return np.allclose(mat, id_mat, rtol=rtol, atol=atol)