Abstract
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 X T AX under the condition X T BX = J 1 where J 1 is a given diagonal matrix of order ≤ n and J 2 1 = I. We also prove an analog of the Cauchy interlacing theorem for some such pencils.
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