Orthostochastic matrix

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

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

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

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

we find the corresponding orthogonal matrix

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

with ${\displaystyle \cos ^{2}\phi =a,}$ such that ${\displaystyle B_{ij}=O_{ij}^{2}.}$

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. Vol. 108. Cambridge: Cambridge University Press. ISBN 0-521-86565-4. Zbl 1106.05001.