Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products

Abstract

We characterized weighted spectral geometric means (SGM) of positive definite matrices in terms of certain matrix equations involving metric geometric means (MGM) ♯ and semi-tensor products ⋉. Indeed, for each real number t and two positive definite matrices A and B of arbitrary sizes, the t-weighted SGM A ⬦t B of A and B is a unique positive solution X of the equation A−1 ♯ X = (A−1 ♯ B)t. We then established fundamental properties of the weighted SGMs based on MGMs. In addition, (A ♢1/2 B)2 is positively similar to A ⋉ B and, thus, they have the same spectrum. Furthermore, we showed that certain equations concerning weighted SGMs and MGMs of positive definite matrices have a unique solution in terms of weighted SGMs. Our results included the classical weighted SGMs of matrices as a special case.