Vol. 49, No. 2, 1973

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Definite and semidefinite matrices in a real symmetric matrix pencil

Frank Uhlig

Vol. 49 (1973), No. 2, 561–568

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 = {(ai,bi)}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 d-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.

Mathematical Subject Classification 2000
Primary: 15A48
Received: 12 July 1972
Published: 1 December 1973
Frank Uhlig