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. It is orthostochastic if there exists an orthogonal matrix O such that
- [math]\displaystyle{ B_{ij}=O_{ij}^2 \text{ for } i,j=1,\dots,n. \, }[/math]
All 2-by-2 doubly stochastic matrices are orthostochastic (and also unistochastic) since for any
- [math]\displaystyle{ B= \begin{bmatrix} a & 1-a \\ 1-a & a \end{bmatrix} }[/math]
we find the corresponding orthogonal matrix
- [math]\displaystyle{ O = \begin{bmatrix} \cos \phi & \sin \phi \\ - \sin \phi & \cos \phi \end{bmatrix}, }[/math]
with [math]\displaystyle{ \cos^2 \phi =a, }[/math] such that [math]\displaystyle{ B_{ij}=O_{ij}^2 . }[/math]
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.
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