# Strict spectral approximation of a matrix and some related problems

Applicationes Mathematicae (1997)

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

## Access Full Article

top## Abstract

top## How to cite

topZiętak, 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

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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.