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

$B_{ij}=O_{ij}^2\text{ for }i,j=1,\dots ,n.$

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

$B=\left[\begin{array}{cc}a& 1-a\\ 1-a& a\end{array}\right]$

we find the corresponding orthogonal matrix

$O=\left[\begin{array}{cc}\cos \varphi & \sin \varphi \\ -\sin \varphi & \cos \varphi \end{array}\right],$

with ${\cos }^2\varphi =a,$ 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.

References

• Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. 108. Cambridge: Cambridge University Press. ISBN 0-521-86565-4. https://archive.org/details/combinatorialmat0000brua.