Abstract :
An n × n real matrix A is TP (totally positive) if all its minors are positive or zero; NTP, if it is nonsingular and TP; STP, if it is strictly TP; O (oscillatory), if it is TP and a power Am is STP. Let P be one of NTP, O, STP. We prove that if A is symmetric and has property P, μ is not an eigenvalue of A, and A - μI = QR and A′ - μI = RQ with R having positive diagonal, then A′ has property P, and vice versa. The analysis includes a new criterion for A to be STP.
