TRACE MINIMIZATION AND DEFINITENESS OF SYMMETRICAL PENCILS

A symmetric matrix pencil A - lambda B of order n is called positive d efinite if there is a mu such that the matrix A - mu B is positive def inite. We consider the case with B nonsingular and show that the defin iteness is closely related to the existence of min Tr X(T) AX under th e condition X(T) BX = J(1) where J(1) is a given diagonal matrix of or der less than or equal to n and J(I)(2) = I. We also prove an analog o f the Cauchy interlacing theorem for some such pencils.