LINEAR COMBINATIONS OF HERMITIAN AND REAL SYMMETRIC MATRICES

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′=(x1,x2, ...,xn)≠(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. © 1979.