Sign patterns that require repeated eigenvalues
An sign-pattern matrix is an \(n\)-by-\(n\) matrix \(B\) whose entries are elements of the set \(\{+,-,0\}\). The sign-pattern class \(Q(B)\) associated with \(B\) is the set of all \(n\)-by-\(n\) real matrices\(A\) for which the entry of \(A\) in the intersection of row \(i\) and column \(j\) is positive (respectively, negative, zero) if the entry of \(B\) in the same location is + (respectively, \(-\), 0). For every positive integer \(k\), the authors characterize all matrices \(B\) for which \(k\) is the smallest integer with the property that every matrix \(A\) in \(Q(B)\) has an eigenvalue of multiplicity at least \(k\). The characterization is given in terms of the size of zero submatrices of \(B\), or in terms of the length of cycles in the directed graph on vertices \(\{1,\dots,n\}\) associated with \(B\) in the usual way. The graph contains an arrow starting at \(i\) and ending at \(j\) if and only if the element of \(B\) in the intersection of row \(i\) and column \(j\) is not 0. The authors discuss the relationship of the above characterization to diagonalizability (by similarity) of matrices in \(Q(B)\) and give several instructive examples.
scientific article; zbMATH DE number 428213