Linear combinations of Hermitian and real symmetric matrices

Abstract This paper, by purely algebraic and elementary methods, studies useful criteria under which the quadratic forms x ′ Ax and x ′ Bx , where A , B are n × n symmetric real matrices and x ′=( x 1 , x 2 , …, x n )≠(0,0,0,0, …,0), can vanish simultaneously and some real linear combination of A , B can be positive definite. Analogous results for hermitian matrices have also been discussed. We have given sufficient conditions on m real symmetric matrices so that some real linear combination of them can be positive definite.

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