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Let $M$ be a row substochastic matrix, with at least one row having sum less than 1. Also, suppose $M$ is irreducible in the sense of a markov chain. Is there an easy way to show the largest eigenvalue must be strictly less than 1? I hope that this result is true as stated. I know that Cauchy interlacing theorem gives me $\leq$,

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    I think you are missing a condition, else you could take $M$ to be the 2 by 2 diagonal matrix with entries 1/2 and 1. The largest eigenvalue of this $M$ is 1.2011-05-04
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    Sorry I meant to add irreducibility. Thanks for the correction2011-05-04

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