Spectrally complete pairs of matrices
A pair (A,B) of \(n\times n\) matrices over a field F is spectrally complete if for every n-tuple \((\lambda_ 1,...,\lambda_ n)\) of elements of F satisfying \(\Sigma \lambda_ i=tr A+tr B\) there is an A' similar to A such that \(A'+B\) has eigenvalues \(\lambda_ 1,...,\lambda_ n\). We denote by i(A) the number of invariant polynomials of A different from 1. The following theorem is the main result: Theorem. Let F be a field, and \(A,B\in F^{n\times n}\), with \(n>2\). Let \(f_ 1(x),...,f_ r(x)(g_ 1(x),...,g_ s(x))\) be the invariant polynomials of A(B) different from 1. (a) If \(i(A)+i(B)\leq n\) and at least one of the following conditions is satisfied: (1) \(n=2\), (2) there is at least one polynomial in the set \(\{f_ 1(x),...,f_ r(x),g_ 1(x),...,g_ s(x)\}\) which does not have degree two, then (A,B) is spectrally complete. (b) The converse of (a) is also true, if F has at least three elements.
scientific article; zbMATH DE number 4063272