秩1矩阵不导致有限性的失效
Rank one matrices do not contribute to the failure of the finiteness property
摘要：
定义d乘d实矩阵或复矩阵的有界集的联合谱半径为从该集合中抽取的矩阵乘积的最大指数增长率。一组矩阵被称为满足有限性，如果这个最大增长率出现在一个周期性的无穷序列上。本文给出了矩阵的有限集满足秩为1的元素的有限性的几个充分条件。我们特别表明，如果一个有限的矩阵集不满足有限性，那么由所有矩阵的秩至少为2的子集是非空的，不满足有限性，并具有相同的联合谱半径的原始集合。我们还得到了至多包含一个非秩一元素的矩阵集合的联合谱半径的一个精确公式，推广了X.戴.
英文摘要：
The joint spectral radius of a bounded set of d times d real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. A set of matrices is said to satisfy the finiteness property if this maximum rate of growth occurs along a periodic infinite sequence. In this note we give some sufficient conditions for a finite set of matrices to satisfy the finiteness property in terms of its rank one elements. We show in particular that if a finite set of matrices does not satisfy the finiteness property, then the subset consisting of all matrices of rank at least two is nonempty, does not satisfy the finiteness property, and has the same joint spectral radius as the original set. We also obtain an exact formula for the joint spectral radii of sets of matrices which contain at most one element not of rank one, generalising a recent result of X. Dai.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
