Vol. 49, No. 2, 1973

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The number of vectors jointly annihilated by two real quadratic forms determines the inertia of matrices in the associated pencil

Frank Uhlig

Vol. 49 (1973), No. 2, 537–542

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 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 l 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 l = n 1.

(II) The pencil P(S,T) contains only indefinite matrices iff n1 l n with other (complementary to the above) conditions holding if l = n1.

Mathematical Subject Classification 2000
Primary: 15A21
Received: 12 July 1972
Published: 1 December 1973
Frank Uhlig