matrix_props.is_totally_positive¶

Checks if the matrix is totally positive.

Functions¶

is_totally_positive(mat[, tol, sub_sizes])

Determine whether a matrix is totally positive. [1].

Module Contents¶

matrix_props.is_totally_positive.is_totally_positive(mat, tol=1e-06, sub_sizes=None)¶

Determine whether a matrix is totally positive. [1].

A totally positive matrix is a square matrix where all the minors are positive. Equivalently, the determinant of every square submatrix is a positive number.

Examples

Consider the matrix

\[\begin{split}X = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\end{split}\]

To determine if this matrix is totally positive, we need to check the positivity of all of its minors. The 1x1 minors are simply the individual entries of the matrix. For \(X\), these are

\[\begin{split}\begin{equation} \begin{aligned} X_{1,1} &= 1 \\ X_{1,2} &= 2 \\ X_{2,1} &= 3 \\ X_{2,2} &= 4 \\ \end{aligned} \end{equation}\end{split}\]

Each of these entries is positive. There is only one 2x2 minor in this case, which is the determinant of the entire matrix \(X\). The determinant of \(X\) is calculated as:

\[\text{det}(X) = 1 \times 4 - 2 \times 3 = 4 - 6 = 2\]

Our function indicates that this matrix is indeed totally positive.

import numpy as np
from toqito.matrix_props import is_totally_positive

A = np.array([[1, 2], [3, 4]])

is_totally_positive(A)

False

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

\[\begin{split}B = \begin{pmatrix} 1 & 2 \\ 3 & -4 \end{pmatrix}\end{split}\]

is not totally positive. The 2x2 minor of \(B\) is the determinant of the entire matrix \(B\). The determinant of \(B\) is calculated as:

\[\text{det}(B) = 1 \times -4 - 2 \times 3 = -4 - 6 = -10\]

Since the determinant is negative, \(B\) is not totally positive.

import numpy as np
from toqito.matrix_props import is_totally_positive

B = np.array([[1, 2], [3, -4]])

is_totally_positive(B)

False

References

Parameters:

mat (numpy.ndarray) – Matrix to check.
tol (float) – The absolute tolerance parameter (default 1e-06).
sub_sizes (list | None) – List of sizes of submatrices to consider. Default is all sizes up to min(mat.shape).

Returns:

Return True if matrix is totally positive, and False otherwise.