Combinatorial Eigenvalues of Matrices

Let S be a subset of diagonal entries of an n × n complex matrix A. When the members of S have a common value which is equal to an eigenvalue of A, then S is a critical diagonal set of A. The existence of such a set is equivalent to the matrix Ã, obtained from A by setting its diagonal entries equal to zero, having an s × t zero submatrix with s + t ≥ n + 1. If S is a minimal critical diagonal set, then equality holds. A zero submatrix of à is used to identify the elements in a critical diagonal set. Then a combinatorial approach is taken in order to study the eigenspace of an eigenvalue associated with a critical diagonal set and to make observations regarding the Frobenius normal form of A and reducibility of A.