Strict spectral approximation of a matrix and some related problems

Krystyna Ziętak

Applicationes Mathematicae (1997)

  • Volume: 24, Issue: 3, page 267-280
  • ISSN: 1233-7234

Abstract

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We show how the strict spectral approximation can be used to obtain characterizations and properties of solutions of some problems in the linear space of matrices. Namely, we deal with (i) approximation problems with singular values preserving functions, (ii) the Moore-Penrose generalized inverse. Some properties of approximation by positive semi-definite matrices are commented.

How to cite

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Ziętak, Krystyna. "Strict spectral approximation of a matrix and some related problems." Applicationes Mathematicae 24.3 (1997): 267-280. <http://eudml.org/doc/219168>.

@article{Ziętak1997,
abstract = {We show how the strict spectral approximation can be used to obtain characterizations and properties of solutions of some problems in the linear space of matrices. Namely, we deal with (i) approximation problems with singular values preserving functions, (ii) the Moore-Penrose generalized inverse. Some properties of approximation by positive semi-definite matrices are commented.},
author = {Ziętak, Krystyna},
journal = {Applicationes Mathematicae},
keywords = {positive semi-definite matrix; $c_p$-minimal approximation; Moore-Penrose generalized inverse; strict spectral approximation of a matrix; singular values preserving functions; -minimal approximation; singular value preserving functions},
language = {eng},
number = {3},
pages = {267-280},
title = {Strict spectral approximation of a matrix and some related problems},
url = {http://eudml.org/doc/219168},
volume = {24},
year = {1997},
}

TY - JOUR
AU - Ziętak, Krystyna
TI - Strict spectral approximation of a matrix and some related problems
JO - Applicationes Mathematicae
PY - 1997
VL - 24
IS - 3
SP - 267
EP - 280
AB - We show how the strict spectral approximation can be used to obtain characterizations and properties of solutions of some problems in the linear space of matrices. Namely, we deal with (i) approximation problems with singular values preserving functions, (ii) the Moore-Penrose generalized inverse. Some properties of approximation by positive semi-definite matrices are commented.
LA - eng
KW - positive semi-definite matrix; $c_p$-minimal approximation; Moore-Penrose generalized inverse; strict spectral approximation of a matrix; singular values preserving functions; -minimal approximation; singular value preserving functions
UR - http://eudml.org/doc/219168
ER -

References

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  1. T. Ando, T. Sekiguchi and T. Suzuki (1973), Approximation by positive operators, Math. Z. 131, 273-282. Zbl0243.47026
  2. F. L. Bauer, J. Stoer and C. Witzgall (1961), Absolute and monotonic norms, Numer. Math. 3, 257-264. Zbl0111.01602
  3. R. Bhatia and F. Kittaneh (1992), Approximation by positive operators, Linear Algebra Appl. 161, 1-9. Zbl0830.47009
  4. R. Bouldin (1973), Positive approximants, Trans. Amer. Math. Soc. 177, 391-403. Zbl0264.47020
  5. C. Davis (1976), An extremal problem for extensions of a sesquilinear form, Linear Algebra Appl. 13, 91-102. Zbl0326.15012
  6. M. Fiedler and T. L. Markham (1993), A characterization of the Moore-Penrose inverse, Linear Algebra Appl. 179, 129-133. Zbl0764.15003
  7. P. E. Gill, W. Murray and M. H. Wright (1981), Practical Optimization, Academic Press, London. Zbl0503.90062
  8. G. H. Golub and C. Van Loan (1989), Matrix Computations, J. Hopkins Univ. Press, Baltimore. Zbl0733.65016
  9. P. R. Halmos (1972), Positive approximants of operators, Indiana Univ. Math. J. 21, 951-960. Zbl0263.47018
  10. N. J. Higham (1989), Matrix nearness problems and applications, in: Application of Matrix Theory, M. J. C. Gover and S. Barnett (eds.), Oxford Univ. Pres, New York, 1-27. 
  11. R. A. Horn and Ch. R. Johnson (1986), Matrix Analysis, Cambridge Univ. Press, Cambridge. 
  12. R. Huotari and W. Li (1994), Continuity of metric projection, Pólya algorithm, strict best approximation, and tubularity of convex sets, J. Math. Anal. Appl. 182, 836-856. Zbl0796.41021
  13. R. E. Kalman (1976), Algebraic aspects of the generalized inverse of a rectangular matrix, in: Generalized Inverses and Applications, M. Z. Nashed (ed.), Academic Press, New York, 111-124. C.-K. Li and N.-K. Tsing (1987), On the unitarily invariant norms and some related results, Linear and Multilinear Algebra 20, 107-119. 
  14. P. J. Maher (1990), Some operator inequalities concerning generalized inverses, Illinois J. Math. 34, 503-514. Zbl0733.47001
  15. R. Penrose (1955), A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51, 406-413. Zbl0065.24603
  16. R. Penrose (1956), On best approximate solutions of linear matrix equations, ibid. 52, 17-19. Zbl0070.12501
  17. C. R. Rao (1973), Linear Statistical Inference and Its Applications, Wiley, New York. Zbl0256.62002
  18. J. R. Rice (1962), Tchebycheff approximation in a compact metric space, Bull. Amer. Math. Soc. 68, 405-410. Zbl0111.26501
  19. D. D. Rogers and J. D. Ward (1981), C p -minimal positive approximants, Acta Sci. Math. (Szeged) 43, 109-115. Zbl0481.47012
  20. E. M. de Sá (1994), Faces of the unit ball of a unitarily invariant norm, Linear Algebra Appl. 197/198, 451-493. Zbl0808.15014
  21. W. So (1990), Facial structures of Schatten p -norms, Linear and Multilinear Algebra 27, 207-212. Zbl0706.15027
  22. G. W. Stewart and J.-G. Sun (1990), Matrix Perturbation Theory, Academic Press, Boston. Zbl0706.65013
  23. R. C. Thompson (1972), Principal submatrices IX: Interlacing inequalities for singular values of submatrices, Linear Algebra Appl. 5, 1-12. Zbl0252.15009
  24. H. J. Woerdeman (1994), Superoptimal completions of triangular matrices, Integral Equations Operator Theory 20, 492-501. Zbl0824.47018
  25. N. J. Young (1986), The Nevanlinna-Pick problem for matrix-valued functions, J. Operator Theory 15, 239-269. Zbl0608.47020
  26. K. Ziętak (1988), On characterization of the extremal points of the unit sphere of matrices, Linear Algebra Appl. 106, 57-75. Zbl0653.15019
  27. K. Ziętak (1993), Properties of linear approximations of matrices in the spectral norm, ibid. 183, 41-60. Zbl0770.15011
  28. K. Ziętak (1995), Strict approximation of matrices, SIAM J. Matrix Anal. Appl. 16, 232-234. Zbl0815.41016

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