Let $D$ be a real diagonal matrix $D=diag(a_1,a_2,\ldots,a_n)$ with $a_1\le a_2\le\ldots\le a_n$. Assume that at least one of the $a_i$ is positive. Let $P$ be an irreducible, real, row-stochastic matrix (all entries between 0 and 1, the sum of the elements of every row is 1), with all its eigenvalues real. Let $k>0$ be a real number. I wish to prove that $D+kP$ has a positive eigenvalue. Do you think this is true? Thanks in advance!
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