Orthostochastic 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, 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

we find the corresponding orthogonal matrix

with cos 2 ⁡ ϕ = a , such that B i j = O i j 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.