Combinatorial eigenvalues of matrices (Q1200561)

Given \(A\in C^{n\times n}\), its spectrum \(\sigma(A)\) and the subset \(S\subseteq\) \{the set of diagonal entries of \(A\)\} one shows that the existence of a critical diagonal \(S\) (i.e. \(S\) and \(\sigma(A)\) have a common element) is equivalent to the matrix \(\tilde A\), obtained from \(A\) by setting its diagonal entries equal to zero, having an \(s\times t\) zero submatrix with \(s+t\geq n+1\) (equality holding when \(S\) is minimal). The study of the eigenspace of an eigenvalue associated with the critical diagonal \(S\), and observations regarding the Frobenius normal form of \(A\) and reducibility of \(A\) by a combinatorial approach are given. Several examples illustrate the theoretical treatment.