THE SPECTRA OF NONNEGATIVE MATRICES VIA SYMBOLIC DYNAMICS

• 成果类型: Article
• 署名作者: BOYLE, M; HANDELMAN, D
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2944339
• 发表日期: 1991
• 页码: 249-316
• 关键词: positive matrices; markov-chains

摘要

We characterize (by elementary conditions) those k-tuples of complex numbers which are the nonzero portion (including multiplicities) of the spectrum of a nonnegative real matrix. The proof relies on methods and results from symbolic dynamics. More generally, let S be a unital subring of the reals. We conjecture that certain elementary necessary conditions are sufficient for a k-tuple DELTA of complex numbers to be the nonzero portion of the spectrum of a primitive matrix over S. (The general nonnegative case would follow easily from the primitive case-but not conversely.) We verify this under the additional condition that some subtuple of DELTA containing its maximal (real) entry be the nonzero portion of the spectrum of a primitive matrix over S. In particular, if the maximal entry of DELTA is in S or is quadratic over it, the elementary necessary conditions are sufficient. As one application (with S = Z), we characterize the zeta functions of mixing finitely presented dynamical systems.