orthostochastic matrix

{{short description|Doubly stochastic matrix}}

In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the squares of

the absolute values of the entries 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. It is orthostochastic if there exists an orthogonal matrix O such that

:$B\_\{ij\}=O\_\{ij\}^2\; \backslash text\{\; for\; \}\; i,j=1,\backslash dots,n.\; \backslash ,$

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

$\backslash cos^2\; \backslash phi\; =a,$ such that

$B\_\{ij\}=O\_\{ij\}^2\; .$

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.