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
Bij
| 2 | |
| =O | |
| ij |
fori,j=1,...,n.
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}
O=\begin{bmatrix} \cos\phi&\sin\phi\\ -\sin\phi&\cos\phi\end{bmatrix},
\cos2\phi=a,
Bij
| 2 | |
| =O | |
| ij |
.
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.