Necessary and sufficient conditions for the positive definiteness and stability of symmetric interval matrices

This paper considers the positive definiteness and stability of symmetric interval matrices. The concepts of variable nonnegative quadratic forms and intermediate separate combination are introduced, and sufficient condition for the positive definiteness of variable nonnegative quadratic forms is derived. They are applied to the positive definiteness problems of symmetric interval matrices so that necessary and sufficient conditions for the positive definiteness of symmetric interval matrices are derived. Based on some properties of positive definite matrices, necessary and sufficient conditions for the stability of symmetric interval matrices are proposed. Some examples are given to demonstrate the applicability of the derived results.