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Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products
Author(s)
Date Issued
January 1, 2024
Type
Article
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.
Citation
Aims Mathematics, 9(5), 11452-11467, 2024