论文标题
在具有独特特征值和独特奇异值的矩阵密集子集上
On dense subsets of matrices with distinct eigenvalues and distinct singular values
论文作者
论文摘要
众所周知，所有$ n \ times n $矩阵的集合具有不同的特征值，这是所有真实或复杂$ n \ times n $矩阵集合的密集子集。在[Hartfiel，D。J.致密矩阵的密集集。 Proc。阿米尔。数学。 Soc。，123（6）：1669-1672，1995。]，作者为所有$ n \ times n $矩阵集的子空间建立了必要且充分的条件，可以具有具有独特特征值的密集矩阵。我们有兴趣为所有$ n \ times n $真实或复杂矩阵的子集找到一些必要和足够的条件，以具有具有不同特征值的密集矩阵子集。我们的一些结果概括了Hartfiel的结果。此外，我们研究了所有真实或复杂矩阵集合的子集中，分别研究具有独特的奇异值，独特的分析特征值和独特的分析奇异值的矩阵的密集子集的存在。
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.
