Definite and semidefinite matrices in a real symmetric matrix pencil

Abstract

Pencils that contain a definite matrix (cZ-pencils) have been characterized in several ways. Here ^-pencils will be characterized by the property of the set L = {(ai9 bi)} ^ R 2 if S and T are simultaneously congruent to diagCa*) and diag(&i), respectively. This way one can describe all definite and semidefinite matrices in a c£-pencil. Similarly one can characterize all pencils that contain semidefinite but no definite matrices (s.d. pencils). The explicit condition on L for cZ-pencils is then used to reprove the theorem that two real symmetric matrices generate a cZ-pencil iff their associated quadratic forms do not vanish simultaneously. DEFINITION 1. If S is symmetric we define Qs = {x e Rn x'Sx = 0}. DEFINITION 2. For real symmetric (r.s.) matrices S and T one defines the pencil P(S, T) = {aS + bT\a, beR}. DEFINITION 3. (a) P(S, T) is called d-pencil if P(S, T) contains a definite matrix. (b) P(S, T) is called s.d. pencil if P(S, T) contains a nonzero