ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

In minisymposium: Geometry and Order Structure in Matrices and Operators

Tue 11:30–12:00, Room AV 03.12
On some results related to positive definite matrices by means of matrix means
Osman Kan (Selcuk University Science Faculty)
Joint work with Ramazan Türkmen (SU Konya)

The matrix $M$= $\left[\begin{array}{cc} A & X \ X^\ast & B% \end{array}% \right]$ is called PPT matrix if the matrices $\left[\begin{array}{cc} A & X \ X^\ast & B% \end{array}% \right]$ and $\left[\begin{array}{cc} A & X^\ast \ X & B% \end{array}% \right]$ are both positive semi definite.

Let A,B be $n\times n$ positive definite matrices. Then arithmetic, geometric and harmonic means of these matrices are $\frac{1}{2}\left(A+B\right)$, $A^{1/2}\left(A^{-1/2}BA^{-1/2}\right)^{1/2}A^{1/2}$ and $2\left(A^{-1}+B^{-1}\right)^{-1}$ respectively.

In this talk we obtain some good results related to positive definite matrices by using properties of PPT matrix, Matrix means and majorization.