New bounds for the Perron root ρ(A)$\rho (A)$$\rho (A)$ of a positive matrix A are proposed. Let R1≥R2≥⋯≥Rn$R_1\ge R_2\ge \cdots \ge R_n$$R_1\ge R_2\ge \cdots \ge R_n$ be the row sums of matrix A$A$ listed in descending order. We use a pairwise combination method to prove that $\underset{1\le k\le j}{min}\{(1-t)R_k+t{R}_{n-k+1}\}\le \rho (A)\le \underset{1\le k\le j}{max}\{t{R}_k+(1-t)R_{n-k+1}\},$with $j=\lceil n/2\rceil$, $t=M/(M+m)$, and M (respectively m) is the maximum (resp. minimum) entry of A. This result improves the well-known bounds $R_n\le \rho (A)\le R_1$, and the same method can also be used to improve several other known bounds.

中文翻译：

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正矩阵 Perron 根的界限

佩伦根的新界限ρ （一）$\rho （A）$$\rho (A)$提出了一个正矩阵A。让 右1≥右2≥ ⋯ ≥右n${右}_1\ge {右}_2\ge \cdots \ge {右}_n$$R_1\ge R_2\ge \cdots \ge R_n$是矩阵的行和A$A$按降序排列。我们用两两组合的方法来证明$\underset{1\le k\le j}{分钟}\{（1-t）{右}_k+t{右}_{n-k+1}\}\le \rho （A）\le \underset{1\le k\le j}{最大限度}\{t{右}_k+（1-t）{右}_{n-k+1}\},$和$j=\lceil n/2\rceil$,$t=\mathrm{中号}/（\mathrm{中号}+米）$，M（分别为m ）是A的最大（或最小）条目。这个结果改善了众所周知的界限${右}_n\le \rho （A）\le {右}_1$，同样的方法也可以用来改进其他几个已知的界限。