In mathematics, an orthostochastic matrix is a

doubly stochastic matrix In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_= ...

whose entries are the squares of the absolute values of the entries of some

orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ...

. 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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

. It is orthostochastic if there exists an

''O'' such that :

B_=O_^2 \text i,j=1,\dots,n. \,

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

unistochastic In mathematics, a unistochastic matrix (also called ''unitary-stochastic'') is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix ''B'' of size ''n'' is doubly sto ...

) since for any :

B= \begin
a & 1-a \\
1-a & a \end

we find the corresponding orthogonal matrix :

O = \begin
\cos \phi &  \sin \phi \\
- \sin \phi & \cos \phi \end,

with

\cos^2 \phi =a,

such that

B_=O_^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

* {{DEFAULTSORT:Orthostochastic Matrix Matrices