Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXTENSION OF BLOCK MATRIX REPRESENTATION OF THE GEOMETRIC MEAN

  • Choi, Hana (Department of Mathematics Sungkyunkwan University) ;
  • Choi, Hayoung (School of Information Science and Technology ShanghaiTech University) ;
  • Kim, Sejong (Department of Mathematics Chungbuk National University) ;
  • Lee, Hosoo (Department of Mathematics Education Teachers College Jeju National University)
  • Received : 2019.04.09
  • Accepted : 2019.07.25
  • Published : 2020.05.01
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Abstract

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

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References

  1. T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979), 203-241. https://doi.org/10.1016/0024-3795(79)90179-4
  2. T. Ando, C.-K. Li, and R. Mathias, Geometric means, Linear Algebra Appl. 385 (2004), 305-334. https://doi.org/10.1016/j.laa.2003.11.019
  3. M. Bakonyi and H. J. Woerdeman, Matrix Completions, Moments, and Sums of Hermitian Squares, Princeton University Press, Princeton, NJ, 2011. https://doi.org/10.1515/9781400840595
  4. R. Bhatia, Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2007.
  5. R. Bhatia and J. Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra Appl. 413 (2006), no. 2-3, 594-618. https://doi.org/10.1016/j.laa.2005.08.025
  6. R. Bhatia and R. L. Karandikar, Monotonicity of the matrix geometric mean, Math. Ann. 353 (2012), no. 4, 1453-1467. https://doi.org/10.1007/s00208-011-0721-9
  7. R. E. Curto and L. A. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17 (1993), no. 2, 202-246. https://doi.org/10.1007/BF01200218
  8. I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, The band method for positive and contractive extension problems, J. Operator Theory 22 (1989), no. 1, 109-155.
  9. I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integral Equations Operator Theory 12 (1989), no. 3, 343-382. https://doi.org/10.1007/BF01235737
  10. I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, The band method for several positive extension problems of nonband type, J. Operator Theory 26 (1991), no. 1, 191-218.
  11. I. Gohberg, M. A. Kaashoek, and H. J. Woerdeman, A maximum entropy principle in the general framework of the band method, J. Funct. Anal. 95 (1991), no. 2, 231-254. https://doi.org/10.1016/0022-1236(91)90029-5
  12. R. Grone, C. R. Johnson, E. M. Sa, and H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984), 109-124. https://doi.org/10.1016/0024-3795(84)90207-6
  13. R. A. Horn and C. R. Johnson, Matrix Analysis, second edition, Cambridge University Press, Cambridge, 2013.
  14. J. D. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly 108 (2001), no. 9, 797-812. https://doi.org/10.2307/2695553
  15. M. O. Omran and W. Barrett, The real positive definite completion problem for a 4-cycle, Linear Algebra Appl. 336 (2001), 131-166. https://doi.org/10.1016/S0024-3795(01)00317-2
  16. W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Mathematical Phys. 8 (1975), no. 2, 159-170. https://doi.org/10.1016/0034-4877(75)90061-0