본문 바로가기
KERD
논문KCI 등재

Extension of block matrix representation of the geometric mean

김세종|이호수|최하나|Hayoung Choi
대한수학회 (The Korean Mathematical Society)|2020.05|대한수학회지 (Journal of the KMS)|제57권 제3호|pp. 641-653|DOI 10.4134/JKMS.j190272

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

Schur complementmatrix geometric meanPositive matrix completion
주제분류수학

초록

To extend the well-known extremal characterization of the geometric mean of two $n \times n$ positive definite matrices $A$ and $B$, we solve the following problem: \begin{equation*} \max \Bigg\{ X:X=X^*,~ \begin{pmatrix} A & V & X \\ V & B & W \\ X & W & C \end{pmatrix} \geq 0 \Bigg \}. \end{equation*} We find an explicit expression of the maximum value with respect to the matrix geometric mean of Schur complements.

Abstract

To extend the well-known extremal characterization of the geometric mean of two $n \times n$ positive definite matrices $A$ and $B$, we solve the following problem: \begin{equation*} \max \Bigg\{ X:X=X^*,~ \begin{pmatrix} A & V & X \\ V & B & W \\ X & W & C \end{pmatrix} \geq 0 \Bigg \}. \end{equation*} We find an explicit expression of the maximum value with respect to the matrix geometric mean of Schur complements.

참고문헌 (16)

  1. T. Ando, "Concavity of certain maps on positive denite matrices and applications to Hadamard products", Linear Algebra Appl, 26, pp.203~241 (1979)[DOI]
  2. T. Ando, "Geometric means", Linear Algebra Appl, 385, pp.305~334 (2004)[DOI]
  3. M. Bakonyi, "Matrix Completions, Moments, and Sums of Her-mitian Squares" (2011), Princeton University Press
  4. R. Bhatia, "Positive Denite Matrices, Princeton Series in Applied Mathematics" (2007), Prince-ton University Press
  5. R. Bhatia, "Riemannian geometry and matrix geometric means", Linear Algebra Appl, 413(2-3), pp.594~618 (2006)[DOI]
  6. R. Bhatia, "Monotonicity of the matrix geometric mean", Math. Ann, 353(4), pp.1453~1467 (2012)[DOI]
  7. R. E. Curto, "Recursively generated weighted shifts and the subnormal completion problem", Integral Equations Operator Theory, 17(2), pp.202~246 (1993)[DOI]
  8. I. Gohberg, "The band method for positive and contractive extension problems", J. Operator Theory, 22(1), pp.109~155 (1989)
  9. I. Gohberg, "The band method for positive and strictly contractive extension problems : an alternative version and new applications", Integral Equations Operator Theory, 12(3), pp.343~382 (1989)[DOI]
  10. I. Gohberg, "The band method for several positive extension problems of nonband type", J. Operator Theory, 26(1), pp.191~218 (1991)
  11. I. Gohberg, "A maximum entropy principle in the general framework of the band method", J. Funct. Anal, 95(2), pp.231~254 (1991)[DOI]
  12. R. Grone, "Positive denite completions of partial Hermitian matrices", Linear Algebra Appl., 58, pp.109~124 (1984)[DOI]
  13. R. A. Horn, "Matrix Analysis" (2013), Cambridge University Press
  14. J. D. Lawson, "The geometric mean, matrices, metrics, and more", Amer. Math. Monthly, 108(9), pp.797~812 (2001)[DOI]
  15. M. O. Omran, "The real positive denite completion problem for a 4-cycle", Linear Algebra Appl, 336, pp.131~166 (2001)[DOI]
  16. W. Pusz, "Functional calculus for sesquilinear forms and the purication map", Rep. Mathematical Phys, 8(2), pp.159~170 (1975)[DOI]

관련 논문AI추천

Positiveness for the Riemannian geodesic block matrix김세정, 황진미대한수학회 · 대한수학회논문집2020 · 제35권 제3호Representations of the Moore-Penrose inverse of 2×2 block operator valued matricesChun Yuan Deng, Hong Ke Du대한수학회 · 대한수학회지2009 · 제46권 제6호Weighted Geometric Means of Positive OperatorsNoboru Nakamura, Saichi Izumino경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2010 · 제50권 제2호Oppenheim and Schur Type Inequalities for Khatri-Rao Products of Positive Definite Matrices김세정, 김정준, 이호수경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2017 · 제57권 제4호Riccati Equation and Positivity of Operator MatricesJun Ichi Fujii, Masatoshi Fujii, Ritsuo Nakamoto경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2009 · 제49권 제4호Geometric Means of Positive OperatorsNoboru Nakamura경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2009 · 제49권 제1호

이 간행물의 다른 논문 (13건)

On complete convergence for extended independent random variables under sub-linear expectationsXuejun Wang, Xin Dengpp. 553-570The Jacobi sums over Galois rings and its absolute values장영호pp. 571-583Construction of recursive formulas generating power moments of Kloosterman sums: $O^{+}(2n,2^{r})$ case김대산pp. 585-602Weighted Hardy inequalities with sharp constantsRyskul Oinarov, Aigerim Kalybaypp. 603-616A parallel iterative method for a finite family of Bregman strongly nonexpansive mappings in reflexive Banach spacesTruong Minh Tuyen, 김종규pp. 617-640
원문 보기