A limit theorem for monotone matrix functions

Abstract

Given a symmetric m × m matrix function Q( t) which decreases on some interval (0, ε], ε > 0 [i.e., Q( t 1)− Q( t 2) is nonnegative definite for t 1⩽ t 2] and which admits a factorization 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 TQ(t)X=U TX , and lim t→0+ c TQ(t)c=∞ for all c ∉ ImX . Moreover, any monotone matrix function can be factorized as above.