Abstract
In this paper we consider the following problem: Given two matricesA,Z∈ℂn×n, does there exist an invertiblen×n-matrixS such thatS −1 AS is an upper triangular matrix andS −1 ZS is a lower triangular matrix, and if so, what can be said about the order in which the eigenvalues ofA andZ appear on the diagonals of these triangular matrices? For special choices ofA andZ a complete solution is possible, as has been shown by several authors. Here we follow a lead, provided by Shmuel Friedland, who discussed the case where bothA andZ have at leastn-1 linearly independent eigenvectors, and we descibe the problem in terms of Jordan chains and left-Jordan chains for the matricesA, Z. The results give some insight in the question why certain classes of matrices (like the nonderogatory and the rank 1 matrices) allow for a detailed solution of the problems described above; for some of these classes the result of this analysis is presented here for the first time.
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Thijsse, P. Spectral criteria for complementary triangular forms. Integr equ oper theory 27, 228–251 (1997). https://doi.org/10.1007/BF01191535
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DOI: https://doi.org/10.1007/BF01191535