Improving bounds for nonmaximal eigenvalues of positive matrices (Q1774965)

Let \(A= [a_{ij}]\) be positive with Perron eigenvalue \(\rho(A)\), and let \(u\), \(v\) be corresponding positive right and left eigenvectors such that \(v^Tu= 1\). Order the eigenvalues of \(A\) so that \(\text{Re}(\lambda_1(A))\leq\cdots\leq \text{Re}(\lambda_{n-1}(A))< \rho(A)\), and define \(\tau(A)= \text{Re}(\lambda_{n-1}(A))\). Bounds on \(\tau(A)\) are important for determining the rate of convergence of powers of \(A\). Suppose \(c\) is a scalar such that \(1+ c> c^*(A)=\rho(A)\max_{i,j}\{{u_i v_j\over a_{ij}}\}\), and define \(A_c= A-{\rho(A)\over 1+c} uv^T\). The key observation underlying the results here is that if \({c\over 1+c}\rho(A)> \tau(A)\), then \(\tau(A_c)= \tau(A)\). Thus if \(\xi(A)\) is any known upper bound for \(\tau(A)\) and additional conditions on \(c\) can be found so that \(\xi(A_c)< \xi(A)\) is true, then \(\xi(A_c)\) is a better bound for \(\tau(A)\). The author is able to do this for the bounds derived by \textit{A. Berman} and \textit{X.-D. Zhang} [Linear Algebra Appl. 316, No. 1--3, 13--20 (2000; Zbl 0958.15014)] and by \textit{R. Nabben} [SIAM J. Matrix Anal. Appl. 22, 574-579 (2000; Zbl 1039.15010)]. He also gives a numerical example which illustrates that the improvement can be substantial.