Definite and semidefinite matrices in a real symmetric matrix pencil

Abstract

Pencils that contain a definite matrix (d-pencils) have been characterized in several ways. Here d-pencils will be charac­terized by the property of the set L = {(αi, bi⊆if R2 if S and T are simultaneously congruent to diag(ai) and diag(bi), respec­tively. This way one can describe all definite and semidefinite matrices in a pencil. Similarly one can characterize all pencils that contain semidefinite but no definite matrices (s.d. pencils). The explicit condition on L for (d-pencils is then used to reprove the theorem that two real symmetric matrices generate a d-pencil iff their associated quadratic forms do not vanish simultaneously. © 1973 by Pacific Journal of Mathematics.