Doubly stochastic matrix

In mathematics refers to a double - stochastic matrix (sometimes double - stochastic transition matrix ) is a square matrix whose row and column sums amount to one and whose elements lie between zero and one.

Characterizations

The following characterizations doubly - stochastic matrices are equivalent:

• A matrix is ​​doubly - stochastic if and only if the row and column sums be one and are all elements of the matrix between zero and one.

• A matrix is doubly stochastic exactly when both the transposed matrix are transition matrices.

• A matrix is ​​doubly - stochastic if and only if the row and column sums amount to one and all elements of the matrix are non-negative.

Eigenvalues ​​and eigenvectors

Like all transition matrices have also doubly stochastic matrices as amount largest eigenvalue of the eigenvalue 1, since each double - stochastic matrix is both row-and spaltenstochastisch, is the one vector ( which only ones as entries has ) both left-and right eigenvector of each double - stochastic matrix. Is now the matrix doubly stochastic and additionally either irreducible or true positive (see Perron - Frobenius theorem of ), so the only stationary distribution of the Markov chain, which is characterized by the uniform distribution, ie the probability vector.

Set of Birkhoff and von Neumann

For a matrix is that it is precisely then double - stochastic if it is a convex combination of permutation matrices.

Additional: The permutation matrices are the extreme points of the set of doubly stochastic matrices.