Geodesic
Classification problems for systems of forms and linear mappings
Izvestiya. Mathematics , Tome 31 (1988) no. 3, pp. 481-501.

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

A method is proposed that allows the reduction of many classification problems of linear algebra to the problem of classifying Hermitian forms. Over the complex, real, and rational fields, classifications are obtained for bilinear forms, pairs of quadratic forms, isometric operators, and selfadjoint operators. Bibliography: 30 titles. 
@article{IM2_1988_31_3_a2,
     author = {V. V. Sergeichuk},
     title = {Classification problems for systems of forms and linear mappings},
     journal = {Izvestiya. Mathematics },
     pages = {481--501},
     publisher = {mathdoc},
     volume = {31},
     number = {3},
     year = {1988},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1988_31_3_a2/}
}
TY  - JOUR
AU  - V. V. Sergeichuk
TI  - Classification problems for systems of forms and linear mappings
JO  - Izvestiya. Mathematics 
PY  - 1988
SP  - 481
EP  - 501
VL  - 31
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1988_31_3_a2/
LA  - en
ID  - IM2_1988_31_3_a2
ER  - 
%0 Journal Article
%A V. V. Sergeichuk
%T Classification problems for systems of forms and linear mappings
%J Izvestiya. Mathematics 
%D 1988
%P 481-501
%V 31
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1988_31_3_a2/
%G en
%F IM2_1988_31_3_a2
V. V. Sergeichuk. Classification problems for systems of forms and linear mappings. Izvestiya. Mathematics , Tome 31 (1988) no. 3, pp. 481-501. http://geodesic.mathdoc.fr/item/IM2_1988_31_3_a2/

[1] Gabriel P., “Underlegbare Darstellungen, I”, Manuscripta Math., 6, 1972, 71–103 | MR | Zbl

[2] Bershtein I. N., Gelfand I. M., Ponomarev V. A., “Funktory Kokstera i teorema Gabrielya”, Uspekhi matem. nauk, 28:2 (1973), 19–33 | MR

[3] Nazarova L. A., “Predstavleniya kolchanov beskonechnogo tipa”, Izv. AN SSSR. Ser. matem., 37:4 (1973), 752–791 | MR | Zbl

[4] Ringel C. M., Tame algebras and integral quadratic forms, Lecture Notes in Math., 1099, Springer, Berlin, 1984 | MR | Zbl

[5] O'Meara O. T., Introduction to quadratic forms, Springer, Berlin, 1971 | MR

[6] Scharlau Winfred. Quadratic and Hermitian forms, Springer, Berlin, 1985 | MR

[7] Gabriel P., “Degenerate bilinear forms”, J. Algebra, 31 (1974), 67–72 | DOI | MR | Zbl

[8] Riehm C., “The equivalence of bilinear forms”, J. Algebra, 31 (1974), 45–66 | DOI | MR | Zbl

[9] Riehm C., Shrader-Frechette M. A., “The equivalence of sesquilinear forms”, J. Algebra, 42 (1976), 495–53 | DOI | MR

[10] Scharlau R., “Zur Klassifikation von Bilinearformen und von Isometrien über Körpern”, Math. Z., 178 (1981), 359–373 | DOI | MR | Zbl

[11] Yaglom I. M., “Kvadratichnye i kososimmetricheskie bilineinye formy v veschestvennykh simplekticheskikh prostranstvakh”, Tr. seminara po vektornomu i tenzornomu analizu, 8, 1950, 364–381 | MR | Zbl

[12] Scharlau R., “Paare alternierender Formen”, Math. Z., 147 (1976), 13–19 | DOI | MR | Zbl

[13] Waterhouse William C., “Pairs of quadratic forms”, Invent. Math., 37:2 (1976), 157–164 | DOI | MR | Zbl

[14] Uhlig Frank, “A recurding theorem about nairs of quadratic forms and extensions a survey”, Linear Algebra and Appl., 25 (1979), 219–237 | DOI | MR | Zbl

[15] Uhlig Frank, “A rational canonical pair form for a pair of symmetric matrices over an arbitrary field $F$ with char $F\ne2$ and applications to finest simultaneous block diagonalizations”, Linear and Multilinear Algebra, 8:1 (1979), 41–67 | DOI | MR | Zbl

[16] Milnor J., “On isometries of inner product spaces”, Invent. Math., 1969, no. 8, 83–97 | DOI | MR | Zbl

[17] Huppert B., “Isometrien von Vektorräumen, I”, Arch. Math. (Basel), 35:1,2 (1980), 164–176 | MR | Zbl

[18] Huppert B., “Isometrien von Vektorräumen, II”, Math. Z., 175:1 (1980), 5–20 | DOI | MR

[19] Lewis D. W., “The isometry classification of Hermitian forms over division algebras”, Linear Algebra and Appl., 43 (1982), 245–272 | DOI | MR | Zbl

[20] Roiter A. V., “Boksy s involyutsiei”, Predstavleniya i kvadratichnye formy, In-t matem. AN USSR, Kiev, 1979, 124–126 | MR

[21] Sergeichuk V. V., “Predstavleniya prostykh involyutivnykh kolchanov”, Predstavleniya i kvadratichnye formy, In-t matem. AN USSR, Kiev, 1979, 127–148 | MR

[22] Kruglyak S. A., Predstavleniya involyutivnykh kolchanov, Dep. v VINITI, No 7266-84, AN USSR. In-t matematiki, Kiev, 1984

[23] Quebbemann H.-G., Scharlau W., Schulie M., “Quadratic and Hermitian forms in additive and Abelian categories”, J. Algebra, 59 (1979), 264–289 | DOI | MR | Zbl

[24] Sergeichuk V. V., “Predstavleniya orskhem”, Lineinaya algebra i teoriya predstavlenii, In-t matem. AN USSR, Kiev, 1983, 110–134 | MR

[25] Sergeichuk V. V., Klassifikatsionnye zadachi dlya sistem lineinykh otobrazhenii i polutoralineinykh form, Dep. v UkrNIINTI, No 196 Uk-D 84, Kievskii un-t, Kiev, 1983

[26] Burbaki N., Algebra (Mnogochleny i polya. Uporyadochennye gruppy), Nauka, M., 1965 | MR

[27] Burbaki N., Algebra (Moduli, koltsa, formy), Nauka, M., 1966 | MR

[28] Dedonne Zh., Geometriya klassicheskikh grupp, Mir, M., 1974 | MR

[29] Bass X., Algebraicheskaya $K$-teoriya, Mir, M., 1973 | MR | Zbl

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