"""Is matrix a normal matrix."""
import numpy as np
from toqito.matrix_props import is_square
[docs]
def is_normal(mat: np.ndarray, rtol: float = 1e-05, atol: float = 1e-08) -> bool:
r"""
Determine if a matrix is normal [WikNormal]_.
A matrix is normal if it commutes with its adjoint
.. math::
\begin{equation}
[X, X^*] = 0,
\end{equation}
or, equivalently if
.. math::
\begin{equation}
X^* X = X X^*
\end{equation}.
Examples
==========
Consider the following matrix
.. math::
A = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
our function indicates that this is indeed a normal matrix.
>>> from toqito.matrix_props import is_normal
>>> import numpy as np
>>> A = np.identity(4)
>>> is_normal(A)
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 normal.
>>> from toqito.matrix_props import is_normal
>>> import numpy as np
>>> B = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> is_normal(B)
False
References
==========
.. [WikNormal] Wikipedia: Normal matrix.
https://en.wikipedia.org/wiki/Normal_matrix
:param mat: The matrix to check.
:param rtol: The relative tolerance parameter (default 1e-05).
:param atol: The absolute tolerance parameter (default 1e-08).
:return: Returns :code:`True` if the matrix is normal and :code:`False` otherwise.
"""
if not is_square(mat):
return False
return np.allclose(mat @ mat.conj().T, mat.conj().T @ mat, rtol=rtol, atol=atol)