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) $ \sharp $ and semi-tensor products $ \ltimes $. Indeed, for each real number $ t $ and two positive definite matrices $ A $ and $ B $ of arbitrary sizes, the $ t $-weighted SGM $ A \, \diamondsuit_t \, B $ of $ A $ and $ B $ is a unique positive solution $ X $ of the equation $ A^{-1}\,\sharp\, X \; = \; (A^{-1}\,\sharp\, B)^t. $ We then established fundamental properties of the weighted SGMs based on MGMs. In addition, $ (A \, \diamondsuit_{1/2} \, B)^2 $ is positively similar to $ A \ltimes 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.

Keywords

• spectral geometric mean
• metric geometric mean
• positive definite matrix
• semi-tensor product