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.
