THE ALLOW SEQUENCE OF DISTINCT EIGENVALUES FOR A SIGN PATTERN

A sign pattern A is a matrix with entries in {+, -, 0}. This article introduces the allow sequence of distinct eigenvalues for an n x n sign pattern A, defined as qseq(A) = (q1, . . . , qn), with qk = 1 if there exists a real matrix with exactly k distinct eigenvalues having pattern A, and qk = 0 otherwise. For example, qseq(A) = (0, . . . , 0, 1) is equivalent to A requiring all distinct eigenvalues, while qseq(A) = (1, 0, . . . , 0) is equivalent to the digraph of A being acyclic. Relationships between the allow sequence for A and composite cycles of the digraph of A are explored to identify zeros in the sequence, while methods based on Jacobian matrices are developed to identify ones in the sequence. When A is an n x n irreducible sign pattern, the possible sequences for qseq(A) are completely determined when n <= 4 and when the sequence has at least n - 4 trailing zeros for n >= 5.