Bounds for the Perron Root of Irreducible Nonnegative Matrices

  •  Ping Liao    


Some bounds for the Perron root $\rho$ of positive matrices are proposed. We proved that$$\max_{1\leq i \leq n}{\bigg(\sum_{k}{a_{ik}m_{ki}}  \bigg)}  \leq \rho \leq \min_{1\leq i \leq n}{\bigg(\sum_{k}{a_{ik}M_{ki}}  \bigg)}.$$where $$m_{ki}=\min_{t}{\frac {(\alpha_k,\beta_t)}{(\alpha_i,\beta_t)}}, M_{ki}=\max_{t}{\frac {(\alpha_k,\beta_t)}{(\alpha_i,\beta_t)}}$$and $\alpha_k$ denotes the $k$-th row of matrix $A$,  $\beta_t$ the $t$-th column of $A$, $(\alpha_k,\beta_t)$ denotes the inner product of $\alpha_k$ and $\beta_t$.And these bounds can also be used to estimate the Perron root of nonnegative irreducible matrices.

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