Minimum number of distinct eigenvalues allowed by a sign pattern
This manuscript deals with the study of the minimum number of distinct eigenvalues allowed by a square sign pattern. A sign pattern is a symbolic matrix \(\mathcal{A}\) with entries \(\{+, -, 0\}\), which represents the combinatorial structure of a real matrix, taking into account the signs of the entries. The qualitative class of a sign pattern \(\mathcal{A}\), denoted by \(Q(\mathcal{A})\), is the set of all real matrices \(A\) such that the signs of their entries coincide with the sign pattern \(\mathcal{A}\). For a matrix \(A \in \mathbb{R}^{n \times n}\), let \(q(A)\) be the number of distinct eigenvalues of \(A\). For a sign pattern \(\mathcal{A}\), \[ q(\mathcal{A})=\min \{ q(A) \ : \ A \in Q(\mathcal{A}) \}. \] The authors study \(q(\mathcal{A})\), give some definitions and determine how the minimum number of distinct eigenvalues changes under certain manipulations of the sign pattern. They also develop some techniques to help determining the number \(q(\mathcal{A})\) and present properties of the directed graphs of \(n \times n\) sign patterns \(\mathcal{A}\) with \(q(\mathcal{A})=1\). The number \(q(\mathcal{A})\) for \(2 \times 2\) and irreducible \(3 \times 3\) sign patterns \(\mathcal{A}\) is determined. The paper ends with an appendix providing a list of all irreducible \(3 \times 3\) sign patterns \(\mathcal{A}\) according to the value of \(q(\mathcal{A})\).
scientific article; zbMATH DE number 7597866
scientific article