Banach Journal of Mathematical Analysis

On a geometric property of positive definite matrices cone

Masatoshi Ito, Yuki Seo, Takeaki Yamazaki, and Masahiro Yanagida

Full-text: Open access


We shall discuss the matrix geometric mean for the positive definite matrices. The set of all $n\times n$ matrices with a suitable inner product will be a Hilbert space, and the matrix geometric mean can be considered as a path between two positive matrices. In this paper, we shall obtain a matrix geometric mean inequality, and as an application of it, a property of Riemannian metric space is given. We also obtain some examples related to our result.

Article information

Banach J. Math. Anal. Volume 3, Number 2 (2009), 64-76.

First available in Project Euclid: 17 December 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A64: Operator means, shorted operators, etc.
Secondary: 47A63: Operator inequalities 47L25: Operator spaces (= matricially normed spaces) [See also 46L07]

Positive matrix Riemannian metric geometric mean


Ito, Masatoshi; Seo, Yuki; Yamazaki, Takeaki; Yanagida, Masahiro. On a geometric property of positive definite matrices cone. Banach J. Math. Anal. 3 (2009), no. 2, 64--76. doi:10.15352/bjma/1261086710.

Export citation