Abstract
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.
Highlights
Let A = be an irreducible nonnegative matrix of order n, the Perron root ρ of A is a positive real eigenvalue of A and any other eigenvalue λ is strictly smaller than ρ in absolute value(|λ| < ρ)
Lemma 2 Let A =n×n be a positive matrix of order n and ρ its Perron root, Suppose x = (x1, x2, ..., xn) is the positive eigenvector of ρ, xi and x j are two arbitrarily elements of x, min a jk ≤ x j ≤ max a jk
Theorem 1 Let A =n×n be a positive matrix of order n and ρ its Perron root, 1≤i≤n aik mki
Summary
Let A = (ai j) be an irreducible nonnegative matrix of order n, the Perron root ρ of A is a positive real eigenvalue of A and any other eigenvalue λ is strictly smaller than ρ in absolute value(|λ| < ρ). For nonnegative matrices with nonzero row sums r1, r2,...,rn, the bounds in (1) were improved by Minc (Minc, 1988) as follows: min i. This result was further generalized by Liu (Shulin Liu,1996). Some new bounds for the Perron root of positive matrices are proposed at first, and these bounds can be used to estimate the Perron root of irreducible nonnegative matrix
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