Trace inequalities and linear programming (with applications to markov chains
Lemod Gurvits · Linear and Multilinear Algebra · 1998
Using a trace inequality for M-matrices prove that where P(row ) stochastic matrix and π is its stationary probabilistic distribution Motivated by this result we introduce a cotion of Π superstochastic matrix ,I,e,a square matrix Q with nonnegative entries is called Π superstochastic iff We study when for a Π superstochastic matrix Q A maong other results we prove that ifΠ1 ≤ Π2≤⋯≤Πn/Π1 ethen the inequality above holds for all Π-superstochastic matrices Q and e is the smallest contant of this type .Our solution is based on a passage to a dual problem of linear programming.We also give alternative linear programming -based proof for the inequality above for stochastic matrices Finally,we prove the following result:ifM+M * is positive definite the eigenvalues of both matrices M −1(M − M :*)and M −2 (M − M :*)have nonnegative real parts.As a direct corollaryu of this result we prove one inqulity above for symmetric matrices Using the idea of this proof we prove entropic inquality for symmetric matrices with nonegative entries