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Universal bounds for positive matrix semigroups

Leo Livshits, Gordon MacDonald, Laurent Marcoux, Heydar Radjavi (2016)

Studia Mathematica

We show that any compact semigroup of positive n × n matrices is similar (via a positive diagonal similarity) to a semigroup bounded by √n. We give examples to show this bound is best possible. We also consider the effect of additional conditions on the semigroup and obtain improved bounds in some cases.

Upper bound for the non-maximal eigenvalues of irreducible nonnegative matrices

Xiao-Dong Zhang, Rong Luo (2002)

Czechoslovak Mathematical Journal

We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.

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