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

Arnon Ploymukda; Kanjanaporn Tansri; Pattrawut Chansangiam; Arnon Ploymukda; Kanjanaporn Tansri; Pattrawut Chansangiam

doi:10.3934/math.2024562

AIMS Mathematics

2024, Volume 9, Issue 5: 11452-11467. doi: 10.3934/math.2024562

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

Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Received: 22 January 2024 Revised: 04 March 2024 Accepted: 07 March 2024 Published: 25 March 2024
MSC : 15A24, 15A69

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.
- spectral geometric mean,
- metric geometric mean,
- positive definite matrix,
- semi-tensor product
Citation: Arnon Ploymukda, Kanjanaporn Tansri, Pattrawut Chansangiam. Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products[J]. AIMS Mathematics, 2024, 9(5): 11452-11467. doi: 10.3934/math.2024562

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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.

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