toqito.matrix_props.is_projection

toqito.matrix_props.is_projection(mat, rtol=1e-05, atol=1e-08)[source]

Check if matrix is a projection matrix [WikProj].

A matrix is a projection matrix if it is positive semidefinite (PSD) and if

\[\begin{equation} X^2 = X \end{equation}\]

where \(X\) is the matrix in question.

Examples

Consider the following matrix

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

our function indicates that this is indeed a projection matrix.

>>> from toqito.matrix_props import is_projection
>>> import numpy as np
>>> A = np.array([[0, 1], [0, 1]])
>>> is_projection(A)
True

Alternatively, the following example matrix \(B\) defined as

\[\begin{split}B = \begin{pmatrix} -1 & -1 \\ -1 & -1 \end{pmatrix}\end{split}\]

is not positive definite.

>>> from toqito.matrix_props import is_projection
>>> import numpy as np
>>> B = np.array([[-1, -1], [-1, -1]])
>>> is_projection(B)
False

References

Parameters:

  • mat – Matrix to check.

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

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

Returns:

Return True if matrix is a projection matrix, and False otherwise.