The spectra of nonnegative integer matrices via formal power series
The spectra of nonnegative integer matrices via formal power series
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通过形式幂级数的非负整数矩阵的谱
DOI：
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发表时间：
2000
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通讯作者：
F. Roush
中科院分区：
文献类型：
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作者：
Ki Hang Kim;N. Ormes;F. Roush
矩阵理论中的一个老问题是确定复数的n元组，它可以作为具有非负元素的矩阵的谱出现(参见[BP94，第4章]或[Min88，第7章])。作者研究了元组由实数组成的情况[Bor95，Cia68，Fri78，Kel71，Per53，Sal72，Sou83，Sul49]，所考虑的矩阵对称的情况[Fie74，JLL96]，以及一般问题[Joh81，LM99，LL79，Rea94，Rea96，Wu w97]。已经给出了各种充要条件和充分条件，但仅对n≤4[≤71，Sul49]实n-字节组和仅对n-Kel3[LL79]复n-字节组已知一个完整的刻画。受符号动力学的启发，博伊尔和汉德尔曼通过做出下面的“光谱猜想”[BH91，BH93](另见[Boy93，§8]和[LM95，第11章])，将注意力重新集中在光谱的非零部分。下面，如果矩阵A的所有项都是非负的，并且对于某个n，A的所有项都是严格正的，则矩阵A是本原的。另外，
An old problem in matrix theory is to determine the n-tuples of complex numbers which can occur as the spectrum of a matrix with nonnegative entries (see [BP94, Chapter 4] or [Min88, Chapter VII]). Authors have studied the case where the ntuple is comprised of real numbers [Bor95, Cia68, Fri78, Kel71, Per53, Sal72, Sou83, Sul49], the case where the matrices under consideration are symmetric [Fie74, JLL96], and the general problem [Joh81, LM99, LL79, Rea94, Rea96, Wuw97]. Various necessary conditions and sufficient conditions have been provided, but a complete characterization is known for real n-tuples only for n ≤ 4 [Kel71, Sul49] and for complex n-tuples only for n ≤ 3 [LL79]. Motivated by symbolic dynamics, Boyle and Handelman refocused attention on the nonzero part of the spectrum by making the following “Spectral Conjecture” [BH91, BH93] (see also [Boy93, §8] and [LM95, Chapter 11]). Below, a matrix A is primitive if all entries of A are nonnegative and for some n, all entries of A are strictly positive. Also,