matrix_props.is_normal

Is matrix a normal matrix.

Module Contents

Functions

is_normal(mat[, rtol, atol])

Determine if a matrix is normal [1].

matrix_props.is_normal.is_normal(mat, rtol=1e-05, atol=1e-08)

Determine if a matrix is normal [1].

A matrix is normal if it commutes with its adjoint

\[\begin{equation} [X, X^*] = 0, \end{equation}\]

or, equivalently if

\[\begin{equation} X^* X = X X^* \end{equation}.\]

Examples

Consider the following matrix

\[\begin{split}A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\end{split}\]

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 \(B\) defined as

\[\begin{split}B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\end{split}\]

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

[1] (1,2)

Wikipedia. Normal matrix. https://en.wikipedia.org/wiki/Normal_matrix.

Parameters:

• mat (numpy.ndarray) – The matrix to check.

• rtol (float) – The relative tolerance parameter (default 1e-05).

• atol (float) – The absolute tolerance parameter (default 1e-08).

Returns:

Returns True if the matrix is normal and False otherwise.

Return type:

bool