Symmetric factorizations and localization of zeroes of rational matrix functions

Linear and Multilinear Algebra

Volumn 40, Issue 3, 1996, Pages 259-281

  Lerer, Leonid ^a   Rodman, Leiba ^b  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b Dept of Economics and Thomas Jefferson Program in Public Policy at the College of William and Mary   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0040206276     PISSN: 03081087     EISSN: None     Source Type: Journal    
DOI: 10.1080/03081089608818444     Document Type: Article

References (54)

1. [AJ] B. D. O. Anderson, E. I. Jury: Generalized Bezoutian and Sylvester matrices in multivariable linear control, IEEE Trans. Automat. Control AC-21 (1976), 551-556.
2. [BGR] J. A Ball, I. Gohberg, L. Rodman: Interpolation of Rational Matrix Functions, OT 45, Birkhäuser Verlag, Basel, 1990.
3. [BKRV] J. A. Ball, J. Kim, L. Rodman, M. Verma: Minimal degree coprime factorizations of rational matrix functions, Linear Algebra Appl. 186 (1993), 117-164.
4. [BRI] J. A. Ball, A. C. M. Ran: Left versus right canonical Wiener-Hopf factorization, Oper. Theory Adv. Appl. 21 (1986), 9-38.
5. [BR2] J. A. Ball, A. C. M. Ran: Optimal Hankel norm model reductions and Wiener-Hopf factorization I: The canonical case, SIAM J. Control Optim. 25 (1987), 362-382.
6. [BGK] H. Bart, I. Gohberg, M. A. Kaashoek: Minimal Factorization of Matrix and Operator Functions, OT1, Birkhäuser, 1979.
7. [BGKV] H. Bart, I. Gohberg, M. A. Kaashoek, P. Van Dooren: Factorizations of transfer functions, SIAM J. Control Optim. 18 (1980), 675-696.
8. [BKAK] R. R. Bitmead, S. Y. Kung, B. D. O. Anderson, T. Kailath: Greatest common divisors via generalized Sylvester and Bezout matrices. IEEE Trans. Automat. Control AC-23 (1978), 1043-1047.
9. [BLW] S. Bittanti, A. J. Laub, J. C. Willems (eds.): The Riccati Equation, Springer Verlag, Berlin, etc., 1991.
10. [CD] D. Carlson, B. M. Datta: On the effective computation of the inertia of a non-hermitian matrix, Numer. Math. 33 (1979), 315-322.
11. [C] C. T. Chen: A generalization of the inertia theorem, SIAM J. Appl. Math. 25 (1993), 158-161.
12. [CK] K. Clancey, B. A. Kon: The Bezoutian and the algebraic Riccati equation, Linear and Multilinear Algebra 15 (1984), 265-278.
13. [Co] W. A. Coppel: Linear Systems, Notes in Pure Mathematics 6 (1972), Australian National University, Canberra.
14. [D] B. N. Datta: On the Routh-Hurwitz-Fujiwara and the Schur-Cohn-Fujiwara Theorems for the root-separation problem, Linear Algebra Appl. 22 (1978), 235-246.
15. [Fuh] P. A. Fuhrmann: On the symmetric rational transfer functions, Linear Algebra Appl. 50 (1983), 167-250.
16. [FO] P. A. Fuhrmann, R. Ober: State space formula for coprime factorizations, Oper. Theory Adv. Appl. OT 62 (T. Faruta, I. Gohberg, T. Nakazi, eds.) 1993, 39-75.
17. [GH] I. Gohberg, G. Heinig: The resultant matrix and its generalizations, II. Continual analog of resultant matrix, Acta Math. Acad. Sci. Hungar. 28 (1976), 198-209 (in Russian).
18. [GLR1] I. Gohberg, P. Lancaster, L. Rodman: Spectral theory of selfadjoint matrix polynomials, Ann. of Math. 112 (1980), 33-71.
19. [GLR2] I. Gohberg, P. Lancaster, L. Rodman: Matrix Polynomials, Academic Press, 1982.
20. [GLR3] I. Gohberg, P. Lancaster, L. Rodman: Matrices and Indefinite Scalar Products, OT 8, Birkhäuser Verlag, 1983.
21. [GLR4] I. Gohberg, P. Lancaster, L. Rodman: Invariant Subspaces of Matrices with Applications, J. Wiley, 1986.
22. [GL] I. Gohberg, L. Lerer: Matrix generalizations of M. G. Krein theorems on orthogonal polynomials, Oper. Theory Adv. Appl. 34, Birkhäuser, Basel (1988), 137-202.
23. [HL1] J. Haimovici, L. Lerer: Bezout operators for analytic operator functions. I. A general concept of Bezout operators, Integral Equations Operator Theory 21 (1995), 33-70.
24. [HL2] J. Haimovici, L. Lerer: Bezout operators for analytic operator functions, II, in preparation.
25. [HR] G. Heinig, K. Rost: Algebraic Methods for Toeplitz-like Matrices and Operators, Birkhäuser, Basel, 1984.
26. [Ja] V. A. Jacubovic: Factorization of symmetric matrix polynomials, Soviet Math. Dokl. 11 (1970), 1261-1264.
27. [K] R. E. Kalman: On the Hermite-Fujiwara Theorem in stability theory, Quart. Appl. Math. 23 (1965), 279-282.
28. [KN] M. G. Krein, M. A. Naimark: The method of symmetric and hermitian forms in theory of separation of the roots of algebraic equations, GNTI, Kharkov, 1936 (Russian); Linear and Multilinear Algebra 10 (1981), 265-308.
29. [LM] P. Lancaster, J. Maroulas: The kernel of the Bezoutian for operator polynomials, Linear and Multilinear Algebra 17 (1985), 181-201.
30. [LaR1] P. Lancaster, L. Rodman: Existence and uniqueness theorems for algebraic Riccati equations, Internat. J. Control 32 (1980), 285-309.
31. [LaR2] P. Lancaster, L. Rodman: Algebraic Riccati Equations, Oxford University Press, 1995.
32. [LaR3] P. Lancaster, L. Rodman: Minimal symmetric factorizations of symmetric real and complex rational matrix functions, Linear Algebra Appl. 220 (1995), 249-282.
33. [LaT] P. Lancaster, M. Tismenetsky: The Theory of Matrices with Applications, 2nd ed., Academic Press, 1985.
34. [L] L. Lerer: The matrix quadratic equations and factorization of matrix polynomials, Oper. Theory Adv. and Appl. 40 (1989), 279-324.
35. [LR1] L. Lerer, L. Rodman: Spectrum separation and inertia for operator polynomials, J. Math. Anal. Appl. 169 (1992), 260-282.
36. [LR2] L. Lerer, L. Rodman: Bezoutians of rational matrix functions, submitted.
37. [LR3] L. Lerer, L. Rodman: Inertia of operator polynomials and stability of differential equations, J. Math. Anal. Appl. 192 (1995), 579-606.
38. [LR4] L. Lerer, L. Rodman: Bezoutians and factorizations of rational matrix functions and matrix equations, Systems and Networks: Mathematical Theory and Applications (U. Helmke, R. Mennicken, J. Saurer, eds.), Akademie Verlag, 1994, pp. 761-766.
39. [LR5] L. Lerer, L. Rodman: Common zero structure of rational matrix functions, J. Funct. Anal., to appear.
40. [LRT1] L. Lerer, L. Rodman, M. Tismenetsky: Bezoutian and the Schur-Cohn problem for operator polynomials, J. Math. Anal. Appl. 103 (1984), 83-102.
41. [LRT2] L. Lerer, L. Rodman, M. Tismenetsky: Inertia theorems for matrix polynomials, Linear and Multilinear Algebra 30 (1991), 157-182.
42. [LT1] L. Lerer, M. Tismenetsky: The Bezoutian and eigenvalue separation problem for matrix polynomials, Integral Equations Operator Theory 5 (1982), 386-445.
43. [LT2] L. Lerer, M. Tismenetsky: Generalized Bezoutian and the inversion problem for block matrices, I. General scheme, Integral Equations Operator Theory 9 (1986), 790-819.
44. [LT3] L. Lerer, M. Tismenetsky: Toeplitz classification of matrices and inversion formulas, II. Block-Toeplitz and perturbed block-Toeplitz matrices, TR 88.197, IBM-Israel Scientific Center, Haifa, August 1986.
45. [LT4] L. Lerer, M. Tismenetsky: Generalized Bezoutian and matrix equations, Linear Algebra Appl. 99 (1988), 123-160.
46. [NS] A. M. Nikolaichuk, I. M. Spitkovsky: On the Riemann boundary-value problem with hermitian matrix, Soviet Math. Dokl. 16 (1975), 533-536.
47. [R] A. C. M. Ran: Minimal factorizations of selfadjoint rational matrix functions, Integral Equations Operator Theory 5 (1982), 850-869.
48. [RR1] A. C. M. Ran, L. Rodman: Factorization of matrix polynomials with symmetries, SIAM J. Matrix Anal. Appl. 15 (1994), 846-864.
49. [RR2] A. C. M. Ran, L. Rodman: Stable invariant lagrangian subspaces: factorization of symmetric rational matrix functions and other applications, Linear Algebra Appl. 137-138 (1990), 575-620.
50. [RR3] A. C. M. Ran, L. Rodman: On symmetric factorizations of rational matrix functions, Linear Multilinear Algebra 29 (1991), 243-261.
51. [S] L. A. Sakhnovich: Operatorial Bezoutian in the theory of separation of roots of entire functions, Funct. Anal. Appl. 10 (1976), 54-61 (in Russian).
52. [Sh] M. Shayman: Geometry of the algebraic Riccati equations, Parts I and II, SIAM J. Control Optim. 21 (1983), 375-394; 395-409.
53. [W] H. K. Wimmer: Inertia theorems for matrices, controllability and linear vibrations, Linear Algebra Appl. 8 (1974), 337-343.
54. [Wo] W. M. Wonham: Linear Multivariable Control: A Geometric Approach, 2-nd edition, Springer Verlag, 1979.