matrix_props.is_projection

Is matrix a projection matrix.

Module Contents

Functions

is_projection(mat[, rtol, atol])

Check if matrix is a projection matrix [1].

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

Check if matrix is a projection matrix [1].

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

[1] (1,2)

Wikipedia. Projection matrix. https://en.wikipedia.org/wiki/Projection_matrix.

Parameters:

• mat (numpy.ndarray) – Matrix to check.

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

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

Returns:

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

Return type:

bool