matrix_props.is_stochastic¶

Checks if the matrix is stochastic.

Functions¶

is_stochastic(mat, mat_type)

Verify matrix is doubly, right or left stochastic.

Module Contents¶

matrix_props.is_stochastic.is_stochastic(mat, mat_type)¶

Verify matrix is doubly, right or left stochastic.

When the nonnegative elements in a row of a square matrix sum up to 1, the matrix is right stochastic and if the columns sum up to 1, the matrix is left stochastic [1].

When a matrix is right and left stochastic, it is a doubly stochastic matrix. [2].

See also

is_doubly_stochastic

Examples

The elements of an identity matrix and a Pauli-X matrix are nonnegative such that the rows and columns sum up to 1. We expect these matrices to be left and right stochastic. The same cannot be said about a Pauli-Z or a Pauli-Y matrix.

>>> import numpy as np
>>> from toqito.matrix_props import is_stochastic
>>> is_stochastic(np.eye(5), "right")
True
>>> is_stochastic(np.eye(5), "left")
True
>>> is_stochastic(np.eye(5), "doubly")
True

>>> from toqito.matrices import pauli
>>> from toqito.matrix_props import is_stochastic
>>> is_stochastic(pauli("X"), "left")
True
>>> is_stochastic(pauli("X"), "right")
True
>>> is_stochastic(pauli("X"), "doubly")
True

>>> from toqito.matrices import pauli
>>> from toqito.matrix_props import is_stochastic
>>> is_stochastic(pauli("Z"), "right")
False
>>> is_stochastic(pauli("Z"), "left")
False
>>> is_stochastic(pauli("Z"), "doubly")
False

References

[1]

Wikipedia. Stochastic matrix. URL: https://en.wikipedia.org/wiki/Stochastic_matrix.

[2]

Wikipedia. Doubly stochastic matrix. URL: https://en.wikipedia.org/wiki/Doubly_stochastic_matrix.

Parameters:

mat (numpy.ndarray) – Matrix of interest
mat_type (str) – Type of stochastic matrix. "left" for left stochastic matrix and "right" for right stochastic matrix and "doubly" for a doubly stochastic matrix.

Returns:

Returns True if the matrix is doubly, right or left stochastic, False otherwise.

Raises:

TypeError – If something other than "doubly", "left" or "right" is used for mat_type

Return type:

bool