Singular value inequalities for the arithmetic, geometric and Heinz means of matrices. (English) Zbl 1242.15015

Gumus, Ibrahim Halil; Hirzallah, Omar; Taskara, Necati

Linear Multilinear Algebra 59, No. 12, 1383-1392 (2011).

Given two complex matrices: \(A\) being positive definite and \(B\) being positive semidefinite, for \(\nu \in [0,1]\), the matrix \((1-\nu)A+\nu B\) is called the \(\nu\)-weighted arithmetic mean of \(A\) and \(B\); and \(A\sharp_\nu B := A^{1/2}( A^{-1/2}B A^{-1/2})^\nu A^{1/2}\) is called the \(\nu\)-weighted geometric mean of \(A\) and \(B\), moreover, \(H_\nu(A,B) := (A\sharp_\nu B + A\sharp_{1-\nu} B)/2\) is called the \(\nu\)-weighted Heinz mean of \(A\) and \(B\). In this paper, several singular value inequalities for matrix expressions involving those means are obtained. Furthermore, some applications of these results are provided.

Reviewer: Edgar Pereira (Covilha)

MSC:

15A42	Inequalities involving eigenvalues and eigenvectors
15B48	Positive matrices and their generalizations; cones of matrices
26E60	Means
47A64	Operator means involving linear operators, shorted linear operators, etc.

Keywords:

arithmetic mean; geometric mean; Heinz mean; positive definite matrix; singular value inequalities

References:

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