Upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices (Q1060267)

This paper is concerned with generalizing results obtained by the reviewer [Explicit forms for ergodicity coefficients and spectrum localization, ibid. 60, 187-197 (1984) and in Linear Multilinear Algebra 14, 343-347 (1983; Zbl 0526.15013)]. Both of these were motivated by earlier papers on stochastic matrices of the second author [J. Appl. Probab. 19, 858-863 (1982; Zbl 0501.60074), and ibid. 20, 277-287 (1983; Zbl 0515.60072)], who himself followed on from a paper of the reviewer [Adv. Appl. Probab. 11, 576-590 (1979; Zbl 0406.60060)]. In the main, the present paper focusses on bounds of coefficient-of-ergodicity type on the absolute value of subdominant eigenvalues of primitive non-negative matrices. The key new result is that for such a matrix such eigenvalues are bounded by \(\max \{\| x^ TA\| |\) \(\| x\| \leq 1\), \(x^ Tw=0\), \(x\in {\mathbb{R}}^ n\}\), where \(w\in {\mathbb{R}}^ n\) is the positive Perron-Frobenius eigenvector of A, and \(\|\) \(\|\) is any norm on \({\mathbb{R}}^ n\). The paper also contains much survey material.