The inverse mean problem of geometric mean and contraharmonic means

Yongdo Lim

doi:10.1016/j.laa.2005.06.013

The inverse mean problem of geometric mean and contraharmonic means

Yongdo Lim

Kyungpook National University

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

In this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices (proposed in [W.N. Anderson, M.E. Mays, T.D. Morley, G.E. Trapp, The contraharmonic mean of HSD matrices, SIAM J. Algebra Disc. Meth. 8 (1987) 674-682])A=X+Y-2(X-1+Y-1)-1,B=X#Y,by proving its equivalence to the well-known nonlinear matrix equation X = T - BX ^-1B where T=12(A+A#(A+8BA-1B)) is the unique positive definite solution of X = A + 2BX^-1B. The inverse mean problem is solvable if and only if B ≤ A.

Original language	English
Pages (from-to)	221-229
Number of pages	9
Journal	Linear Algebra and Its Applications
Volume	408
Issue number	1-3
DOIs	https://doi.org/10.1016/j.laa.2005.06.013
State	Published - 1 Oct 2005
Externally published	Yes

Keywords

Contraharmonic mean
Geometric mean
Inverse mean problem
Nonlinear matrix equation
Positive definite matrix

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10.1016/j.laa.2005.06.013

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title = "The inverse mean problem of geometric mean and contraharmonic means",

abstract = "In this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices (proposed in [W.N. Anderson, M.E. Mays, T.D. Morley, G.E. Trapp, The contraharmonic mean of HSD matrices, SIAM J. Algebra Disc. Meth. 8 (1987) 674-682])A=X+Y-2(X-1+Y-1)-1,B=X\#Y,by proving its equivalence to the well-known nonlinear matrix equation X = T - BX -1B where T=12(A+A\#(A+8BA-1B)) is the unique positive definite solution of X = A + 2BX-1B. The inverse mean problem is solvable if and only if B ≤ A.",

keywords = "Contraharmonic mean, Geometric mean, Inverse mean problem, Nonlinear matrix equation, Positive definite matrix",

author = "Yongdo Lim",

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doi = "10.1016/j.laa.2005.06.013",

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AU - Lim, Yongdo

PY - 2005/10/1

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N2 - In this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices (proposed in [W.N. Anderson, M.E. Mays, T.D. Morley, G.E. Trapp, The contraharmonic mean of HSD matrices, SIAM J. Algebra Disc. Meth. 8 (1987) 674-682])A=X+Y-2(X-1+Y-1)-1,B=X#Y,by proving its equivalence to the well-known nonlinear matrix equation X = T - BX -1B where T=12(A+A#(A+8BA-1B)) is the unique positive definite solution of X = A + 2BX-1B. The inverse mean problem is solvable if and only if B ≤ A.

AB - In this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices (proposed in [W.N. Anderson, M.E. Mays, T.D. Morley, G.E. Trapp, The contraharmonic mean of HSD matrices, SIAM J. Algebra Disc. Meth. 8 (1987) 674-682])A=X+Y-2(X-1+Y-1)-1,B=X#Y,by proving its equivalence to the well-known nonlinear matrix equation X = T - BX -1B where T=12(A+A#(A+8BA-1B)) is the unique positive definite solution of X = A + 2BX-1B. The inverse mean problem is solvable if and only if B ≤ A.

KW - Contraharmonic mean

KW - Geometric mean

KW - Inverse mean problem

KW - Nonlinear matrix equation

KW - Positive definite matrix

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DO - 10.1016/j.laa.2005.06.013

M3 - Article

AN - SCOPUS:24344464847

SN - 0024-3795

VL - 408

SP - 221

EP - 229

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

ER -

The inverse mean problem of geometric mean and contraharmonic means

Abstract

Keywords

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