It is sometimes sufficient to use the matrix equation above and the fact that "'Q "'is a stochastic matrix to solve for "'Q " '.
32.
Thus, a doubly stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each row and the sum of the entries in each column is 1.
33.
He obtained a tightening of the Birkhoff von Neumann theorem with H . K . Farahat stating that every doubly stochastic matrix can be obtained as a convex combination of spectra of doubly stochastic matrices.
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I just want to understand Stochastic matrix as fast as possible, so I can grasp some cool formulas published regarding GCL . : )-- talk ) 07 : 44, 26 January 2008 ( UTC)
35.
As left and right eigenvalues of a square matrix are the same, every stochastic matrix has, at least, a row eigenvector associated to the eigenvalue 1 and the largest absolute value of all its eigenvalues is also 1.
36.
However, for a matrix with strictly positive entries ( or, more generally, for an irreducible aperiodic stochastic matrix ), this vector is unique and can be computed by observing that for any i we have the following limit,
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Examples include a stochastic matrix, which describes a stochastic process known as a Markov process, and stochastic calculus, which involves differential equations and integrals based on stochastic processes such as the Wiener process, also called the Brownian motion process.
38.
When a stochastic matrix,, acts on a column vector, " " ", the result is a column vector whose entries are affine combinations of " " " with coefficients from the rows in.
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To satisfy the constraints, it is possible to use a result due to Sinkhorn, which states that a doubly stochastic matrix is obtained from any square matrix with all positive entries by the iterative process of alternating row and column normalizations.
40.
In particular, the state of a probabilistic automaton is always a stochastic vector, since the product of any two stochastic matrices is a stochastic matrix, and the product of a stochastic vector and a stochastic matrix is again a stochastic vector.
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