具有不同特征值和不同奇异值的矩阵的密集子集

Himadri Lal Das, M. Kannan
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引用次数: 0

摘要

众所周知,所有具有不同特征值的$ n \乘以n $矩阵的集合是所有实或复$ n \乘以n $矩阵的集合的密集子集。在[hartfield, D. j]中,可对角化矩阵的密集集。Proc,阿米尔。数学。Soc。中国生物医学工程学报,32(6):1669-1672,1995。],建立了所有$n \ × n$矩阵集合的子空间具有具有不同特征值的矩阵的稠密子集的充分必要条件。我们感兴趣的是找到一些必要和充分条件,使得所有n × n的实矩阵或复矩阵的集合的子集有一个具有不同特征值的矩阵的稠密子集。我们的一些结果推广了哈特菲尔的结果。此外,我们还研究了在所有实矩阵或复矩阵集合的子集中,具有不同奇异值矩阵、不同解析特征值矩阵和不同解析奇异值矩阵的密集子集的存在性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the dense subsets of matrices with distinct eigenvalues and distinct singular values
It is well known that the set of all $ n \times n $ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $ n \times n $ matrices. In [Hartfiel, D. J. Dense sets of diagonalizable matrices. Proc. Amer. Math. Soc., 123(6): 1669-1672, 1995.], the author established a necessary and sufficient condition for a subspace of the set of all $n \times n$ matrices to have a dense subset of matrices with distinct eigenvalues. We are interested in finding a few necessary and sufficient conditions for a subset of the set of all $n \times n$ real or complex matrices to have a dense subset of matrices with distinct eigenvalues. Some of our results are generalizing the results of Hartfiel. Also, we study the existence of dense subsets of matrices with distinct singular values, distinct analytic eigenvalues, and distinct analytic singular values, respectively, in the subsets of the set of all real or complex matrices.
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