In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an orthostochastic matrix is a doubly stochastic matrix

Doubly stochastic matrix

In mathematics, especially in probability and combinatorics, a doubly stochastic matrix,is a square matrix of nonnegative real numbers, each of whose rows and columns sums to 1...

 whose entries are the square of
the absolute value of some orthogonal matrix

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

.

The detailed definition is as follows. An 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 number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, each of whose rows and columns sums to 1. It is orthostochastic if there exists an orthogonal matrix

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

 O such that


All 2-by-2 doubly stochastic matrices are orthostochastic (and also unistochastic

Unistochastic matrix

In mathematics, a unistochastic matrix is a doubly stochastic matrix whose entries are the square of the absolute value of some unitary matrix.The detailed definition is as follows...

)
since for any
we find the corresponding orthogonal matrix
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.