Unistochastic matrix
Encyclopedia
In mathematics
, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix
whose entries are the square of the absolute value of some unitary 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 number
s, each of whose rows and columns sums to 1. It is unistochastic if there exists a unitary matrix U such that
All 2-by-2 doubly stochastic matrices are unistochastic and orthostochastic
, but for larger n it is not the case.
Already for
there exist a bistochastic matrix B which is not unistochastic:
since any two vectors with moduli equal to the square root of the entries of two columns (rows)
of B cannot be made orthogonal by a suitable choice of phases.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix
Doubly stochastic matrix
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix,is a square matrix of nonnegative real numbers, each of whose rows and columns sums to 1...
whose entries are the square of the absolute value of some unitary 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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, each of whose rows and columns sums to 1. It is unistochastic if there exists a unitary matrix U such that
All 2-by-2 doubly stochastic matrices are unistochastic and orthostochastic
Orthostochastic matrix
In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the square ofthe absolute value of some orthogonal matrix.The detailed definition is as follows...
, but for larger n it is not the case.
Already for
there exist a bistochastic matrix B which is not unistochastic:since any two vectors with moduli equal to the square root of the entries of two columns (rows)
of B cannot be made orthogonal by a suitable choice of phases.
Properties
- the set of unistochastic matrices contains all permutation matricesPermutation matrixIn mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
- for
this set is not convex - for
the set of unistochastic matrices is star shaped. - for
the relative volume of the set of unistochastic matrices with respect to the Birkhoff polytope of bistochastic matricesDoubly stochastic matrixIn mathematics, especially in probability and combinatorics, a doubly stochastic matrix,is a square matrix of nonnegative real numbers, each of whose rows and columns sums to 1...
is