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1997 | 38 | 1 | 443-452
Tytuł artykułu

Complementary triangular forms

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The notion of simultaneous reduction of pairs of matrices and linear operators to triangular forms is introduced and a survey of known material on the subject is given. Further, some open problems are pointed out throughout the text. The paper is meant to be accessible to the non-specialist and does not contain any new results or proofs.
Słowa kluczowe
Rocznik
Tom
38
Numer
1
Strony
443-452
Opis fizyczny
Daty
wydano
1997
Twórcy
autor
  • Center for Mathematics and Informatics, P.O. Box 94079, NL-1090 GB, The Netherlands
Bibliografia
  • [1] N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. 60 (1954), 345-350.
  • [2] H. Bart, Transfer functions and operator theory, Linear Algebra Appl. 84 (1986), 33-61.
  • [3] H. Bart, I. Gohberg and M. A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, Oper. Theory: Adv. Appl. 1, Birkhäuser, Basel, 1979.
  • [4] H. Bart and H. Hoogland, Complementary triangular forms of pairs of matrices, realizations with prescribed main matrices, and complete factorization of rational matrix functions, Linear Algebra Appl. 103 (1988), 193-228.
  • [5] H. Bart and L. G. Kroon, Companion based matrix functions: description and minimal factorization, Linear Algebra Appl. 248 (1996), 1-46.
  • [6] H. Bart and L. G. Kroon, Factorization and job scheduling: a connection via companion based rational matrix functions, ibid., to appear.
  • [7] H. Bart and L. G. Kroon, Variants of the two machine flow shop problem, European J. Oper. Res., to appear.
  • [8] H. Bart and G. Ph. A. Thijsse, Complementary triangular forms of upper triangular Toeplitz matrices, in: Oper. Theory: Adv. Appl. 40, Birkhäuser, 1989, 133-149.
  • [9] H. Bart and G. Ph. A. Thijsse, Complementary triangular forms of nonderogatory, Jordan and rank one matrices, Report 9003/B, Econometric Institute, Erasmus University Rotterdam, 1990.
  • [10] H. Bart and G. Ph. A. Thijsse, Eigenspace and Jordan-chain techniques for the description of complementary triangular forms, Report 9353/B, Econometric Institute, Erasmus University Rotterdam, 1993.
  • [11] H. Bart and H. K. Wimmer, Simultaneous reduction to triangular and companion forms of pairs of matrices: the case rank(I-AZ) = 1, Linear Algebra Appl. 150 (1991), 443-461.
  • [12] H. Bart and R. A. Zuidwijk, Triangular forms after extensions with zeroes, submitted.
  • [13] M. P. Drazin, J. W. Dungey and K. W. Gruenberg, Some theorems on commutative matrices, J. London Math. Soc. 26 (1951), 221-228.
  • [14] P. Enflo, A counterexample to the approximation property in Banach spaces, Acta Math. 130 (1973), 309-317.
  • [15] S. Friedland, Pairs of matrices which do not admit a complementary triangular form, Linear Algebra Appl. 150 (1990), 119-123.
  • [16] G. Frobenius, Über vertauschbare Matrizen, Sitz.-Ber. Akad. Wiss. Berlin 26 (1896), 601-614.
  • [17] F. J. Gaines and R. C. Thompson, Sets of nearly triangular matrices, Duke Math. J. 35 (1968), 441-453.
  • [18] I. Gohberg and M. G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, Transl. Math. Monographs 24, A.M.S, Providence, R.I., 1969.
  • [19] T. J. Laffey, Simultaneous triangularization of a pair of matrices, J. Algebra 44 (1977), 550-557.
  • [20] T. J. Laffey, Simultaneous triangularization of matrices--low rank cases and the nonderogatory case, Linear and Multilinear Algebra 6 (1978), 269-305.
  • [21] T. J. Laffey, Simultaneous reduction of sets of matrices under similarity, Linear Algebra Appl. 84 (1986), 123-138.
  • [22] C. Laurie, E. Nordgren, H. Radjavi and P. Rosenthal, On triangularization of algebras of operators, J. Reine Angew. Math. 327 (1981), 143-155.
  • [23] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Second Edition with Applications, Academic Press, Orlando, Fla., 1985.
  • [24] N. H. McCoy, On the characteristic roots of matric polynomials, Bull. Amer. Math. Soc. 42 (1936), 592-600.
  • [25] G. J. Murphy, Triangularizable algebras of compact operators, Proc. Amer. Math. Soc. 84 (1982), 354-356.
  • [26] H. Radjavi, A trace condition equivalent to simultaneous triangularizability, Canad. J. Math. 38 (1986), 376-386.
  • [27] J. R. Ringrose, Non-Self-Adjoint Compact Linear Operators, van Nostrand, New York, 1971.
  • [28] S. H. Tan and J. Vandewalle, On factorizations of rational matrices, IEEE Trans. Circuits and Systems 35 (1988), 1179-1181.
  • [29] R. A. Zuidwijk, Complementary triangular forms for pairs of matrices and operators, doctoral thesis, 1994.
  • [30] R. A. Zuidwijk, Quasicomplete factorizations for rational matrix functions, Integral Equations Operator Theory, to appear.
  • [31] R. A. Zuidwijk, H. Bart and L. Kroon, Quasicomplete factorization and the two machine flow shop problem, Report 9632/B, Econometric Institute, Erasmus University Rotterdam, 1996.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv38i1p443bwm
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