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It has been shown that the geometric mean $A \# B$ of positive definite Hermitian matrices $A$ and $B$ is the maximal element $X$ of Hermitian matrices such that \begin{displaymath} \left( \begin{array}{cc} A & X \\ X & B \\ \end{array} \right) \end{displaymath} is positive semi-definite. As an extension of this result for the $2 \times 2$ block matrix, we consider in this article the block matrix $[[ A \#_{w_{ij}} B ]]$ whose $(i,j)$ block is given by the Riemannian geodesics of positive definite Hermitian matrices $A$ and $B$, where $w_{ij} \in \mathbb{R}$ for all $1 \leq i, j \leq m$. Under certain assumption of the Loewner order for $A$ and $B$, we establish the equivalent condition for the parameter matrix $\omega = [w_{ij}]$ such that the block matrix $[[ A \#_{w_{ij}} B ]]$ is positive semi-definite.
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참고문헌 (7)
참고문헌 신청
T. Ando / 1979 / Concavity of certain maps on positive definite matrices and applications to Hadamard products / Linear Algebra Appl 26:203~241
R. Bhatia / 2007 / Positive Definite Matrices, Princeton Series in Applied Mathematics / Princeton University Press
R. E. Curto / 1993 / Recursively generated weighted shifts and the subnormal completion problem / Integral Equations Operator Theory 17(2):202~246
R. A. Horn / 2013 / Matrix Analysis / Cambridge University Press
S. Kim / 2019 / Geometric mean block matrices / Linear Algebra Appl 575:299~313
R. Bhatia / 2007 / Positive Definite Matrices, Princeton Series in Applied Mathematics / Princeton University Press
R. E. Curto / 1993 / Recursively generated weighted shifts and the subnormal completion problem / Integral Equations Operator Theory 17(2):202~246
R. A. Horn / 2013 / Matrix Analysis / Cambridge University Press
S. Kim / 2019 / Geometric mean block matrices / Linear Algebra Appl 575:299~313
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