|
|||||
|
|
|
|
|
|
|
|
|
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
|
Abstract |
|
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 n− 1 ≦ l ≦ n with other (complementary to the above) conditions holding if l = n− 1. |
Mathematical Subject Classification 2000
Primary: 15A21
|
Milestones
Received: 12 July 1972
Published: 1 December 1973
|
Authors |
| Frank Uhlig | |
|
