Flanders’ theorem for many matrices under commutativity assumptions

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Fernando De Terán
Universidad Carlos III de Madrid

Given two matrices $A \in C^{m \times n}$ and $B \in C^{n \times m}$ , it is known [1] that the Jordan canonical form of AB and BA can only differ in the sizes of the Jordan blocks associated with the eigenvalue zero, and the difference in the size of any two corresponding blocks is, at most, 1. Moreover, this change
is exhaustive, in the sense that given two ordered lists of numbers such that the corresponding elements differ at most by 1, then there are two matrices A, B such that the fist list is the list of sizes of Jordan blocks associated with 0 in AB and the second one is the list of sizes of Jordan blocks associated with 0 in BA.

The motivation of this work is the question: What happens with products of more than two matrices? First, we will see that, if we do not impose any restriction, the products ABC and CBA may have completely different eigenvalues. Therefore, some conditions must be imposed to the factors in order to be able to relate the Jordan canonical forms of any two products.

By imposing some natural commutativity assumptions, we analyze the relationship between the Jordan canonical form of products, in different orders, of k square matrices $A_1, ..., A_k$. In particular, we study permuted products of $A_1, ..., A_k$ under the assumption that the graph of non-commutativity relations 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.

This is joint work with Ross A. Lippert, Yuji Nakatsukasa, and Vanni Noferini.