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

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.
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[1] DOI: 10.1016/0024-3795(94)90341-7 · Zbl 0798.15024 · doi:10.1016/0024-3795(94)90341-7
[2] Bhatia R, Graduate Text in Mathematics, Vol. 169 (1997)
[3] DOI: 10.1016/S0024-3795(00)00048-3 · Zbl 0974.15016 · doi:10.1016/S0024-3795(00)00048-3
[4] Bhatia R, Linear Algebra Appl. 428 pp 2177– (2008) · Zbl 1148.15014 · doi:10.1016/j.laa.2007.11.030
[5] DOI: 10.1007/BF01371042 · Zbl 0412.47013 · doi:10.1007/BF01371042
[6] DOI: 10.1090/S0002-9939-1972-0285556-7 · doi:10.1090/S0002-9939-1972-0285556-7
[7] Mitrinovic DS, Analytic Inequalities (1970) · Zbl 0199.38101 · doi:10.1007/978-3-642-99970-3
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