Finding a positive definite linear combination of two Hermitian matrices

Let A and B be n-by-n Hermitian matrices over the complex field. A result of Au-Yeung 1] and Stewart 8] states that if

$\text{x}{}^1\text{(A + iB)x≠0}$

for all nonzero n-vectors x, then there is a linear combination of A and B which is positive definite. In this article we present an algorithm which finds such a linear combination in a finite number of steps. We also discuss the implementation of the algorithm in case A and B are real symmetric sparse matrices.