Geodesic
Weighted pseudoinverses and weighted normal pseudosolutions with singular weights
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 8, pp. 1347-1363 
E. F. Galba; V. S. Deineka; I. V. Sergienko. Weighted pseudoinverses and weighted normal pseudosolutions with singular weights. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 8, pp. 1347-1363. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_8_a0/
@article{ZVMMF_2009_49_8_a0,
     author = {E. F. Galba and V. S. Deineka and I. V. Sergienko},
     title = {Weighted pseudoinverses and weighted normal pseudosolutions with singular weights},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1347--1363},
     year = {2009},
     volume = {49},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_8_a0/}
}
TY  - JOUR
AU  - E. F. Galba
AU  - V. S. Deineka
AU  - I. V. Sergienko
TI  - Weighted pseudoinverses and weighted normal pseudosolutions with singular weights
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2009
SP  - 1347
EP  - 1363
VL  - 49
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_8_a0/
LA  - ru
ID  - ZVMMF_2009_49_8_a0
ER  - 
%0 Journal Article
%A E. F. Galba
%A V. S. Deineka
%A I. V. Sergienko
%T Weighted pseudoinverses and weighted normal pseudosolutions with singular weights
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2009
%P 1347-1363
%V 49
%N 8
%U http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_8_a0/
%G ru
%F ZVMMF_2009_49_8_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

Weighted pseudoinverses with singular weights can be defined by a system of matrix equations. For one of such definitions, necessary and sufficient conditions are given for the corresponding system to have a unique solution. Representations of the pseudoinverses in terms of the characteristic polynomials of symmetrizable and symmetric matrices, as well as their expansions in matrix power series or power products, are obtained. A relationship is found between the weighted pseudoinverses and the weighted normal pseudosolutions, and iterative methods for calculating both pseudoinverses and pseudosolutions are constructed. The properties of the weighted pseudoinverses with singular weights are shown to extend the corresponding properties of weighted pseudoinverses with positive definite weights.

[1] Ward J. F., Bouillon T. L., Lewis T. O., “Weighted pseudoinverses with singular weights”, SIAM J. Appl. Math., 21:3 (1971), 480–482 | DOI | MR | Zbl

[2] Galba E. F., Deineka B. C., Sergienko I. V., “Razlozheniya i mnogochlennye predelnye predstavleniya vzveshennykh psevdoobratnykh matrits”, Zh. vychisl. matem. i matem. fiz., 47:5 (2007), 747–766 | MR

[3] Sergienko I. V., Galba E. F., Deineka B. C., “Predstavleniya i razlozheniya vzveshennykh psevdoobratnykh matrits, iteratsionnye metody i regulyarizatsiya zadach. 2. Vyrozhdennye vesa”, Kibernetika i sistemnyi analiz, 2008, no. 3, 75–102 | MR | Zbl

[4] Sergienko I. V., Galba E. F., Deineka B. C., “Predstavleniya i razlozheniya vzveshennykh psevdoobratnykh matrits, iteratsionnye metody i regulyarizatsiya zadach. 1. Polozhitelno-opredelennye vesa”, Kibernetika i sistemnyi analiz, 2008, no. 1, 47–73 | MR | Zbl

[5] Chipman J. S., “On least squares with insufficient obserwation”, J. Amer. Statist. Assoc., 59:308 (1964), 1078–1111 | DOI | MR | Zbl

[6] Moore E. H., “On the reciprocal of the general algebraic matrix”, Abstract. Bull. Amer. Math. Soc., 26 (1920), 394–395

[7] Penrose R., “A generalized inverse for matrices”, Proc. Cambridge Philos. Soc., 51:3 (1955), 406–413 | DOI | MR | Zbl

[8] Albert A., Regressiya, psevdoinversiya i rekurrentnoe otsenivanie, Nauka, M., 1975 | MR

[9] Galba E. F., Molchanov H. H., Skopetskii V. V., “Iteratsionnye metody dlya vychisleniya vzveshennoi psevdoobratnoi matritsy s vyrozhdennymi vesami”, Kibernetika i sistemnyi analiz, 1999, no. 5, 150–169 | MR | Zbl

[10] Galba1 E. F., Deineka B. C., Sergienko I. V., “Predelnye predstavleniya vzveshennykh psevdoobratnykh matrits s vyrozhdennymi vesami i regulyarizatsiya zadach”, Zh. vychisl. matem. i matem. fiz., 44:11 (2004), 1928–1946 | MR | Zbl

[11] Galba E. F., “Iteratsionnye metody dlya vychisleniya vzveshennogo normalnogo psevdoresheniya s vyrozhdennymi vesami”, Zh. vychisl. matem. i matem. fiz., 39:6 (1999), 882–896 | MR | Zbl

[12] Lancaster P., Rozsa P., “Eigenvectors of $H$-self-adjoint matrices”, Z. Angew. Math. und Mech., 64:9 (1984), 439–441 | DOI | MR | Zbl

[13] Ikramov Kh. D., “Ob algebraicheskikh svoistvakh klassov psevdoperestanovochnykh i $H$-samosopryazhennykh matrits”, Zh. vychisl. matem. i matem. fiz., 32:8 (1992), 1155–1169 | MR | Zbl

[14] Gantmakher F. R., Teoriya matrits, Nauka, M., 1967 | MR

[15] Sergienko I. V., Galba E. F., Deineka B. C., “Predelnye predstavleniya vzveshennykh psevdoobratnykh matrits s polozhitelno-opredelennymi vesami i regulyarizatsiya zadach”, Kibernetika i sistemnyi analiz, 2003, no. 6, 46–65 | MR | Zbl

[16] Decell H. P., “An application of the Cayley-Hamilton theorem to generalized matrix inversion”, SIAM Rev., 1:4 (1965), 526–528 | DOI | MR

[17] Tikhonov A. H., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1974 | MR

[18] Eiden L., “A weighted pseudoinverse generalized singular values and constrained least squares problems”, BIT, 22:4 (1982), 487–502 | DOI | MR

[19] Morozov B. A., Regulyarnye metody resheniya nekorrektno postavlennykh zadach, Nauka, M., 1987 | MR

[20] Arkharov E. V., Shafiev P. A., “Metody regulyarizatsii zadachi svyazannogo psevdoobrascheniya s priblizhennymi dannymi”, Zh. vychisl. matem. i matem. fiz., 43:3 (2003), 347–353 | MR | Zbl

[21] Ikramov Kh. D., Matin far M., “O kompyuterno-algebraicheskikh protsedurakh dlya lineinoi zadachi naimenshikh kvadratov s lineinymi svyazyami”, Zh. vychisl. matem. i matem. fiz., 44:2 (2004), 206–212 | MR | Zbl

[22] Khorn R., Dzhonson Ch., Matrichnyi analiz, Mir, M., 1989 | MR

[23] Sergienko I. V., Galba E. F., Deineka B. C., “Razlozhenie vzveshennykh psevdoobratnykh matrits v matrichnye stepennye proizvedeniya”, Ukr. matem. zhurnal, 56:11 (2004), 1539–1556 | MR | Zbl