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.
@article{bwmeta1.element.bwnjournal-article-zmv24i3p267bwm, author = {Krystyna Zi\k etak}, title = {Strict spectral approximation of a matrix and some related problems}, journal = {Applicationes Mathematicae}, volume = {24}, year = {1997}, pages = {267-280}, zbl = {0885.15016}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv24i3p267bwm} }
Ziętak, Krystyna. Strict spectral approximation of a matrix and some related problems. Applicationes Mathematicae, Tome 24 (1997) pp. 267-280. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv24i3p267bwm/
[000] T. Ando, T. Sekiguchi and T. Suzuki (1973), Approximation by positive operators, Math. Z. 131, 273-282. | Zbl 0243.47026
[001] F. L. Bauer, J. Stoer and C. Witzgall (1961), Absolute and monotonic norms, Numer. Math. 3, 257-264. | Zbl 0111.01602
[002] R. Bhatia and F. Kittaneh (1992), Approximation by positive operators, Linear Algebra Appl. 161, 1-9. | Zbl 0830.47009
[003] R. Bouldin (1973), Positive approximants, Trans. Amer. Math. Soc. 177, 391-403. | Zbl 0264.47020
[004] C. Davis (1976), An extremal problem for extensions of a sesquilinear form, Linear Algebra Appl. 13, 91-102. | Zbl 0326.15012
[005] M. Fiedler and T. L. Markham (1993), A characterization of the Moore-Penrose inverse, Linear Algebra Appl. 179, 129-133. | Zbl 0764.15003
[006] P. E. Gill, W. Murray and M. H. Wright (1981), Practical Optimization, Academic Press, London. | Zbl 0503.90062
[007] G. H. Golub and C. Van Loan (1989), Matrix Computations, J. Hopkins Univ. Press, Baltimore. | Zbl 0733.65016
[008] P. R. Halmos (1972), Positive approximants of operators, Indiana Univ. Math. J. 21, 951-960. | Zbl 0263.47018
[009] 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.
[010] R. A. Horn and Ch. R. Johnson (1986), Matrix Analysis, Cambridge Univ. Press, Cambridge.
[011] 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. | Zbl 0796.41021
[012] 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.
[013] P. J. Maher (1990), Some operator inequalities concerning generalized inverses, Illinois J. Math. 34, 503-514. | Zbl 0733.47001
[014] R. Penrose (1955), A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51, 406-413. | Zbl 0065.24603
[015] R. Penrose (1956), On best approximate solutions of linear matrix equations, ibid. 52, 17-19. | Zbl 0070.12501
[016] C. R. Rao (1973), Linear Statistical Inference and Its Applications, Wiley, New York. | Zbl 0256.62002
[017] J. R. Rice (1962), Tchebycheff approximation in a compact metric space, Bull. Amer. Math. Soc. 68, 405-410. | Zbl 0111.26501
[018] D. D. Rogers and J. D. Ward (1981), -minimal positive approximants, Acta Sci. Math. (Szeged) 43, 109-115. | Zbl 0481.47012
[019] E. M. de Sá (1994), Faces of the unit ball of a unitarily invariant norm, Linear Algebra Appl. 197/198, 451-493. | Zbl 0808.15014
[020] W. So (1990), Facial structures of Schatten -norms, Linear and Multilinear Algebra 27, 207-212. | Zbl 0706.15027
[021] G. W. Stewart and J.-G. Sun (1990), Matrix Perturbation Theory, Academic Press, Boston. | Zbl 0706.65013
[022] R. C. Thompson (1972), Principal submatrices IX: Interlacing inequalities for singular values of submatrices, Linear Algebra Appl. 5, 1-12. | Zbl 0252.15009
[023] H. J. Woerdeman (1994), Superoptimal completions of triangular matrices, Integral Equations Operator Theory 20, 492-501. | Zbl 0824.47018
[024] N. J. Young (1986), The Nevanlinna-Pick problem for matrix-valued functions, J. Operator Theory 15, 239-269. | Zbl 0608.47020
[025] K. Ziętak (1988), On characterization of the extremal points of the unit sphere of matrices, Linear Algebra Appl. 106, 57-75. | Zbl 0653.15019
[026] K. Ziętak (1993), Properties of linear approximations of matrices in the spectral norm, ibid. 183, 41-60. | Zbl 0770.15011
[027] K. Ziętak (1995), Strict approximation of matrices, SIAM J. Matrix Anal. Appl. 16, 232-234. | Zbl 0815.41016