Let $A\in[0,1]^{n\times n}$ be row stochastic matrix. Let $\gamma \in(0,1)$ be arbitrary. Then how can I prove that $(I-\gamma A)^{-1}$ (where $I$ is the identity matrix) exists?
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If $(I- \gamma A)x=0$, then $Ax= \frac{1}{\gamma}x.$ Now suppose that $x \ne 0$, then $\frac{1}{\gamma}$ is an eigenvalue of $A$. Since $A$ is a row stochastic matrix, $\frac{1}{\gamma} \le 1,$ a contradiction. Thus $x=0.$ Hence $I- \gamma A$ is invertible.
Fred
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How can I prove that the spectral radius of A is one? – shashank ranjan Aug 12 '22 at 10:44
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@shashankranjan Check this out: https://math.stackexchange.com/questions/40320/proof-that-the-largest-eigenvalue-of-a-stochastic-matrix-is-1 – user773458 Aug 12 '22 at 10:53