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Given that matrix $A$ is set as:
$A=(I-P)(I-QP)^{-1}$
where matrix $QP$ is non-negative reducible hollow matrix, and $\rho(QP)<1$. Matrix $P$ is a diagonal matrix and all entries in $P$ are in interval $[0,1]$
I am trying to proof that the spectral radius $\rho(A)<1$. I used Neumann Series to rewrite the matrix $(I-QP)^{-1}$ as
$\sum_{k=0}^{\infty}(QP)^k$, then I am stuck here. How could I proof that the spectral radius of matrix $A$ is less than $1$? Any help will be appreciated!

Simon Liu
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    Which definition of a hollow matrix are you using? Wikipedia gives a few different ones here. – Carl Schildkraut Jul 15 '21 at 05:03
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    I use this definition "square matrix whose diagonal elements are all equal to zero". – Simon Liu Jul 15 '21 at 12:38
  • there's a lot that seems off in this post. I think you mean "irreducible" not "reducible" -- there's little reason to consider the latter. You also must mean $\sum_{k=0}^{\infty}(QP)^k$ for the Neumann series. Let $P:= \frac{1}{2}I$ and let $Q:= \alpha(J-I)$ where $J$ is the one's matrix. With suitable choice of positive $\alpha$ we can make the spectral radius of $QP = \lambda$, the Perron root, arbitrarily close to $1$. Then $(I-QP)^{-1}$ has Perron root $\frac{1}{1-\lambda}$ and $A$ has spectral radius $\frac{1}{2(1-\lambda)}\gt 1$ for $\lambda$ close enough to 1 – user8675309 Jul 15 '21 at 22:55
  • Thanks for your suggestion. I have edited the typo in the Neumann series. But the matrix QP is reducible, and another condition is all entries in P are in interval [0,1]. – Simon Liu Jul 16 '21 at 01:35
  • (1) My counter-example still works. (2) reducibles decompose (i.e. reduce) into blocks of irreducibles and hence for spectral purposes it is both necessary and sufficient for your claim to work on irreducibles. It fails, see (1). Focusing on a reducible matrix is a distraction. Any decent book on markov chains will take this approach. If you don't understand this you can still take my example and stick it in a block diagonal matrix as one of the blocks, then get a counter-example. – user8675309 Jul 17 '21 at 16:43

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