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2013.20: Flanders' theorem for many matrices under commutativity assumptions

2013.20: Fernando De Teran, Ross Lippert, Yuji Nakatsukasa and Vanni Noferini (2013) Flanders' theorem for many matrices under commutativity assumptions.

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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 non-commutativity 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 dierence 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.

Item Type:MIMS Preprint
Uncontrolled Keywords:eigenvalue, Jordan canonical form, Segre characteristic, product of matrices, permuted products, Flanders' theorem, forest, cut- ip.
Subjects:MSC 2000 > 05 Combinatorics
MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2013.20
Deposited By:Yuji Nakatsukasa
Deposited On:26 November 2013

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