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A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. A substochastic matrix is a real square matrix whose row sums are all ; In the same vein, one may define a probability vector as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right ...
The class of doubly stochastic matrices is a convex polytope known as the Birkhoff polytope.Using the matrix entries as Cartesian coordinates, it lies in an ()-dimensional affine subspace of -dimensional Euclidean space defined by independent linear constraints specifying that the row and column sums all equal 1.
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
In 1926, Van der Waerden conjectured that the minimum permanent among all n × n doubly stochastic matrices is n!/n n, achieved by the matrix for which all entries are equal to 1/n. [18] Proofs of this conjecture were published in 1980 by B. Gyires [ 19 ] and in 1981 by G. P. Egorychev [ 20 ] and D. I. Falikman; [ 21 ] Egorychev's proof is an ...
If A is an n × n matrix with strictly positive elements, then there exist diagonal matrices D 1 and D 2 with strictly positive diagonal elements such that D 1 AD 2 is doubly stochastic. The matrices D 1 and D 2 are unique modulo multiplying the first matrix by a positive number and dividing the second one by the same number. [1] [2]
A matrix is called scaled-bistochastic if all elements are non-negative, and the sum of each row and column equals c, where c is some positive constant. In other words, it is c times a bistochastic matrix. Since the positivity graph is not affected by scaling: The positivity graph of any scaled-bistochastic matrix admits a perfect matching.
Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the ...
A Markov chain can be described by a stochastic matrix, which lists the probabilities of moving to each state from any individual state. From this matrix, the probability of being in a particular state n steps in the future can be calculated. A Markov chain's state space can be partitioned into communicating classes that describe which states ...