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Inverse multiparameter eigenvalue problems for matrices III

This note will complement and, in a certain sense, complete our earlier studies [3, 4] of the theory of inverse multiparameter eigenvalue problems for matrices. In those papers, we considered the so called “additive inverse problem” which, briefly stated for the 2-parameter case, asks for conditions on given n × n matrices A, B, C and on given points (si, ti) ∈ ℝ2, 1 ≦ i ≦ n, under which a diagonal matrix D can be found so that the 2-parameter eigenvalue problem can be solved when (λ,μ)=(si, ti), 1 = i = n. Put another way, we look for conditions ensuring that the points (si, ti), 1 ≦ i ≦ n, belong to the eigenvalues of (1.1).