@article{Ploymukda2024, 
author = {Arnon Ploymukda and Kanjanaporn Tansri and Pattrawut Chansangiam},
title = {Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {5},
pages = {11452-11467},
keywords = {spectral geometric mean, metric geometric mean, positive definite matrix, semi-tensor product},
url = {https://www.sciopen.com/article/10.3934/math.2024562},
doi = {10.3934/math.2024562},
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.}
}