zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A characterization and representation of the generalized inverse $A_{T,S}^{(2)}$ and its applications. (English) Zbl 0934.15003
Summary: This paper presents an explicit expression for the generalized inverse $A^{(2)}_{T,S}$. Based on this, we established the characterization, the representation theorem and the limiting process for $A^{(2)}_{T,S}$. As an application, we estimate the error bound of the iterative method for approximating $A^{(2)}_{T,S}$.

MSC:
 15A09 Matrix inversion, generalized inverses 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 65F10 Iterative methods for linear systems
Full Text:
References:
 [1] Ben-Israel, A.: On matrices of index zero or one. SIAM J. Appl. math. 17, 1118-1125 (1968) · Zbl 0186.34002 [2] Ben-Israel, A.; Greville, T. N. E.: Generalized inverses: theory and applications. (1974) · Zbl 0305.15001 [3] Bott, R.; Duffin, R. J.: On the algebra of networks. Trans. amer. Math. soc. 74, 99-109 (1953) · Zbl 0050.25104 [4] Campbell, S. L.; Meyer Jr., C. D.: Generalized inverses of linear transformations. (1979) · Zbl 0417.15002 [5] Carlson, D.; Haynsworth, E.; Markham, T. L.: A generalization of the Schur complement by means of the Moore-Penrose inverse. SIAM J. Appl. math. 26, 169-175 (1974) · Zbl 0245.15002 [6] Chen, Y.: The generalized Bott-Duffin inverse and its applications. Linear algebra appl. 134, 71-91 (1990) · Zbl 0703.15006 [7] Cline, R. E.; Greville, T. N. E.: A Drazin inverse for rectangular matrices. Linear algebra appl. 29, 53-62 (1980) · Zbl 0433.15002 [8] Eldén, L.: A weighted pseudoinverse, generalized singular values and constrained least squares problems. Bit 22, 487-502 (1982) · Zbl 0509.65019 [9] Fiedler, M.; Markham, T. L.: A characterization of the Moore-Penrose inverse. Linear algebra appl. 179, 129-133 (1993) · Zbl 0764.15003 [10] Getson, A. J.; Hsuan, F. C.: [2]-inverses and their statistical applications. Lecture notes in statistics 47 (1988) · Zbl 0671.62003 [11] Groetsch, C. W.: Representations of the generalized inverse. J. math. Anal. appl. 49, 154-157 (1975) · Zbl 0295.47012 [12] Gulliksson, M.: Iterative refinement for constrained and weighted linear least squares. Bit 34, 239-253 (1994) · Zbl 0808.65040 [13] Hanke, M.; Neumann, M.: Preconditionings and splittings for rectangular systems. Numer. math. 57, 85-95 (1990) · Zbl 0704.65018 [14] Husen, F.; Langenberg, P.; Getson, A.: The [2]-inverse with applications to satistics. Linear algebra appl. 70, 241-248 (1985) · Zbl 0584.62078 [15] Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices. Linear and multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003 [16] Mitra, S. K.; Hartwig, R. E.: Partial orders based on outer inverse. Linear algebra appl. 176, 3-20 (1992) · Zbl 0778.15003 [17] Nashed, M. Z.: Generalized inverse and applications. (1976) · Zbl 0428.34008 [18] Nashed, M. Z.: Inner, outer, and generalized inverses in Banach and Hilbert spaces. Numer. funct. Anal. optim. 9, 261-325 (1987) · Zbl 0633.47001 [19] Nashed, M. Z.; Chen, X.: Convergence of Newton-like methods for singular operator equations using outer inverses. Numer. math. 66, 235-257 (1993) · Zbl 0797.65047 [20] Rao, C. R.; Mitra, S. K.: Generalized inverse of matrices and its applications. (1971) · Zbl 0236.15004 [21] Wedin, P. -å.: Perturbation results and condition numbers for outer inverses and especially for projections. Technical report UMINF 124.85, S-901 87 (1985) [22] Wei, M.; Zhang, B.: Structures and the uniqueness conditions of MK-weighted pseudoinverse. Bit 34, 437-450 (1994) · Zbl 0811.15003 [23] Wang, G.; Wei, Y.: Limiting expression for generalized inverse $A(2)$T,S and its corresponding projectors. Numer. math. J. chinese univ. 4, 25-30 (1995) · Zbl 0847.65025 [24] Wei, Y.: A characterization and representation of the Drazin inverse. SIAM J. Matrix anal. Appl. 17, 744-747 (1996) · Zbl 0872.15006 [25] Wei, Y.; Wang, G.: The perturbation theory for the Drazin inverse and its applications. Linear algebra appl. 258, 179-186 (1997) · Zbl 0882.15003 [26] Wei, Y.: Perturbation and computation of the generalized inverse $A(2)$T,S. Master thesis (1994) [27] Wei, Y.: Solving singular linear systems and generalized inverse. Ph.d. thesis (1997) [28] Wei, Y.; Wang, G.: A survey on the generalized inverse $A(2)$T,S. Proceedings of meeting on matrix analysis and applications, 421-428 (1997)