Trace minimization and definiteness of symmetric pencils Original Research Article

A symmetric matrix pencil A - λB of order n is called positive definite if there is a μ such that the matrix A − μB is positive definite. We consider the case with B nonsingular and show that the definiteness is closely related to the existence of min Tr XTAX under the condition XT BX = J1 where J1 is a given diagonal matrix of order ≤ n and J21 = I. We also prove an analog of the Cauchy interlacing theorem for some such pencils.