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Let $P$ be a square matrix and $ \rho(P)$ the spectral radius of $P$. How to show \begin{align} \rho\left( \dfrac{P}{ \rho(P) + \epsilon } \right) < 1 \text{ for all } \epsilon >0. \end{align}

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    Start by thinking about the relationship between the spectrum of $P$ and the spectrum of $\lambda P$, for $\lambda$ a scalar.2012-02-09

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