INVARIANT NEUTRAL SUBSPACES FOR SYMMETRICAL AND SKEW REAL MATRIX PAIRS

Real matrix pairs (A, H) satisfying det H not-equal 0, H(T) = xiH, and HA = etaA(T)H, where xi, eta take the values + 1 or - 1, are considered. It is shown that maximal A-invariant H-neutral subspaces have the same dimension (depending on xi and eta), called the order of neutrality of the pair (A, H). The order of neutrality of definitizable pairs is investigated. In particular, this concept is used to obtain lower bounds for the number of pure imaginary eigenvalues of low rank perturbations of definitizable pairs when (xi, eta) = (1, - 1) and when (xi, eta) = (- 1, - 1).