Abstract
In this note, we prove the equalities for the weighted geometric mean of two accretive matrices A and B: $$\begin{aligned} A\sharp _\nu B=B\sharp _{1-\nu }A=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{\nu }A^{\frac{1}{2}}=B^{\frac{1}{2}}(B^{-\frac{1}{2}}AB^{-\frac{1}{2}})^{1-\nu }B^{\frac{1}{2}},\quad 0<\mathrm{Re\,}\nu <1, \end{aligned}$$ which inherit the same expressions as positive semidefinite matrices. We also prove some weighted AM–GM–HM inequalities for such accretive matrices.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.