SAMS Subject Classification Number: 11

The nonnegative inverse eigenvalue problem asks for conditions on a set of \(n\) complex numbers for it to be the set of eigenvalues of an \(n\times n\) entrywise nonnegative matrix. The problem is known to be very difficult in the generality in which it is stated here: solutions exist for \(n=3\) and \(n=4\) only, along with a list of necessary conditions for the higher dimensional cases. In addition, for the case \(n=5\) a complete solution exists under the extra assumption that the trace of the matrix is zero.

Motivated by the latter observation, we considered for \(n=3\) all possible zero patterns in the matrix, and investigated what extra conditions on the set of eigenvalues these give. In addition, some results for \(n=4\) were obtained as well. In the talk we aim to provide a bit of the flavor of the problem and the techniques used for the solution.