Flanders%26apos; theorem for many matrices under commutativity assumptions

作者：De Teran Fernando^*; Lippert Ross A; Nakatsukasa Yuji; Noferini Vanni

来源：Linear Algebra and Its Applications, 2014, 443: 120-138.

DOI：10.1016/j.laa.2013.11.026

• Eigenvalue
• Jordan canonical form
• Segre characteristic
• Product of matrices
• Permuted products
• Flanders%26apos
• theorem
• Forest
• Cut-flip

We analyze the relationship between the Jordan canonical form of products, in different orders, of k square matrices A(1),..., A(k). Our results extend some classical results by H. Flanders. Motivated by a generalization of Fiedler matrices, we study permuted products of A(1,)..., A(k) under the assumption that the graph of noncommutativity relations of A(1),...., A(k) is a forest. Under this condition, we show that the Jordan structure of all nonzero eigenvalues is the same for all permuted products. For the eigenvalue zero, we obtain an upper bound on the difference between the sizes of Jordan blocks for any two permuted products, and we show that this bound is attainable. For k = 3 we show that, moreover, the bound is exhaustive.

• 出版日期2014-2-15