Positive definite combination of symmetric matrices

This correspondence presents a new algorithm to find a positive definite combination of some given real symmetric matrices. When this problem has solutions, we show that it can be written as an optimization problem. An iterative algorithm using a relaxation type method is proposed to solve this optimization problem. We also prove that this algorithm has a finite-step global convergence property. A comparison, by computer simulations, shows that our algorithm is much faster than another one generally proposed in literature to find such a combination.