A LIMIT-THEOREM FOR MONOTONE MATRIX FUNCTIONS

Given a symmetric m X m matrix function Q(t) which decreases on some i nterval (0, epsilon], epsilon > 0 [i.e., Q(t1) - Q(t2) is nonnegative definite for t1 less-than-or-equal-to t2] and which admits a factoriza tion of the form Q(t) = U(t)X-1(t), where U(t) --> U, X(t) = X as t -- > 0 + with rank(U(T), X(T)) = m. Then it is shown that lim(t --> 0+) X (T)Q(t)X = U(T)X, and lim(t --> 0+) c(T)Q(t)c = infinity for all c is- not-an-element-of Im X. Moreover, any monotone matrix function can be factorized as above.