Orthostochastic Matrix

In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the square of the absolute value of some orthogonal matrix.

The detailed definition is as follows. A square matrix B of size n is doubly stochastic (or bistochastic) if all its rows and columns sum to 1 and all its entries are nonnegative real numbers, each of whose rows and columns sums to 1. It is orthostochastic if there exists an orthogonal matrix O such that

All 2-by-2 doubly stochastic matrices are orthostochastic (and also unistochastic) since for any

B= \begin{bmatrix} a & 1-a \\ 1-a & a \end{bmatrix}

we find the corresponding orthogonal matrix

O = \begin{bmatrix} \cos \phi & \sin \phi \\ - \sin \phi & \cos \phi \end{bmatrix},

with such that

For larger n the sets of bistochastic matrices includes the set of unistochastic matrices, which includes the set of orthostochastic matrices and these inclusion relations are proper.

Famous quotes containing the word matrix:

    ““The matrix is God?”
    “In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
    “If it has limits, it isn’t omnipotent.”
    “Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””
    —William Gibson (b. 1948)