Bounds for the Perron Root of Irreducible Nonnegative Matrices

• DOI: 10.5539/jmr.v1n1p97
• Publication Date: 2015-02-23T21:55:14Z
• Authors: Ping Liao
• Keywords: BETA (programming language); Root (linguistics); Matrix (chemical analysis)

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