The number of vectors jointly annihilated by two real quadratic forms determines the inertia of matrices in the associated pencil
Frank Uhlig · Pacific Journal of Mathematics · 1973
Pencils of real symmetric matrices and their associated quadratic forms are interrelated.It is well known that a pencil contains a definite matrix iff the associated quadratic forms do not vanish simultaneously, provided the matrices have dimension n Ξ> 3.This knowledge is extended here to yield the following for nonsingular pairs of real symmetric matrices of dimension n ^ 3:(I) The pencil P(S, T) contains a semidefinite, but no definite matrix iff the maximal number I of lin.ind.vectors simultaneously annihilated by the associated quadratic forms lies between 1 and n -1 and certain conditions on S and T hold if I = n -1.(II) The pencil P(S, T) contains only indefinite matrices iff n -1 <>l ^ n with other (complementary to the above) conditions holding if I = n -1.