How and why to solve the operator equation AX - XB = Y

Bulletin of the London Mathematical Society

Volumn 29, Issue 1, 1997, Pages 1-21

  Bhatia, Rajendra ^a   Rosenthal, Peter ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0003365357     PISSN: 00246093     EISSN: None     Source Type: Journal    
DOI: 10.1112/S0024609396001828     Document Type: Article

References (104)

1. A. A. ABRAMOV, 'Numerical solution of certain algebraic problems arising in the theory of stability', Akad. Nauk SSSR Zhurnal Vychislitelnoi Mat. Matematicheskoi Fiziki 24 (1984) 339-347.
2. C. A. AKEMANN and P. A. OSTRAND, 'The spectrum of a derivation of a C* algebra', J. London Math. Soc. (2) 13 (1976) 525-530.
3. W. ARENDT, F. RÄBIGER and A. SOUROUR, 'Spectral properties of the operator equation AX + XB = Y', Quart. J. Math. Oxford Ser. (2) 45 (1994) 133-149.
4. J. B. BARLOW, M. M. MONAHEMI and D. P. O'LEARY, 'Constrained matrix Sylvester equations', SIAM J. Matrix Anal. Appl. 13 (1992) 1-9.
5. R. H. BARTELS and G. W. STEWART, 'Algorithm 432: solution of the matrix equation AX + XB = C', Comm. ACM 15 (1972) 820-826.
6. R. BELLMAN, Introduction to matrix analysis (2nd edn, McGraw-Hill, New York, 1970).
7. H. BERCOVICI, 'Notes on invariant subspaces', Bull. Amer. Math. Soc. (N.S.) 23 (1990) 1-33.
8. R. BHATIA, Perturbation bounds for matrix eigenvalues (Longman, Harlow, 1987).
9. R. BHATIA, 'Matrix factorizations and their perturbations', Linear Algebra Appl. 197/198 (1994) 245-276.
10. R. BHATIA, 'First and second order perturbation bounds for the operator absolute value', Linear Algebra Appl. 208/209 (1994) 367-376.
11. R. BHATIA, C. DAVIS and P. KOOSIS, 'An extremal problem in Fourier analysis with applications to operator theory', J. Funct. Anal. 82 (1989) 138-150.
12. R. BHATIA, C. DAVIS and A. MCINTOSH, 'Perturbation of spectral subspaces and solution of linear operator equations', Linear Algebra Appl. 52/53 (1983) 45-67.
13. A. BROWN and C. PEARCY, 'Structure of commutators of operators', Ann. Math. 82 (1965) 112-127.
14. T = C', IEEE Trans. Automat. Control AC-29 (1984) 926-928.
15. JU. L. DALECKI, 'On the asymptotic solution of a vector differential equation', Dokl. Akad. Nauk SSSR 92 (1953) 881-884 (Russian).
16. JU. L. DALECKI and M. G. KREIN, Stability of solutions of differential equations in Banach space, Transl. Math. Monographs 43 (Amer. Math. Soc., Providence, RI, 1974).
17. K. DATTA, 'The matrix equation XA - BX = R and its applications', Linear Algebra Appl. 109 (1988) 91-105.
18. C. DAVIS and W. KAHAN, 'The rotation of eigenvectors by a perturbation III', SIAM J. Numer. Anal. 7 (1970) 1-46.
19. C. DAVIS and P. ROSENTHAL, 'Solving linear operator equations', Canad. J. Math. XXVI (1974) 1384-1389.
20. E. DE SOUZA and S. P. BHATTACHARYYA, 'Controllability, observability and the solution of AX - XB = C', Linear Algebra Appl. 39 (1981) 167-188.
21. A. S. DEIF, 'Error bounds for the equation AX + XB = C', Matrix Tensor Quart. 34 (1983/84) 66-74.
22. L. DIECI, 'Some numerical considerations and Newton's method revisited for solving algebraic Riccati equations', IEEE Trans. Automat. Control 36 (1991) 608-616.
23. T. E. DJAFERIS and S. K. MITTER, 'Algebraic methods for the study of some linear matrix equations', Linear Algebra Appl. 44 (1982) 125-142.
24. N. DUNFORD, 'Spectral operators', Pacific J. Math. 4 (1954) 321-354.
25. M. R. EMBRY, 'A connection between commutativity and separation of spectra of linear operators', Acta Sci. Math. (Szeged) 32 (1971) 235-237.
26. M. R. EMBRY and M. ROSENBLUM, 'Spectra, tensor products, and linear operator equations', Pacific J. Math. 53 (1974) 95-107.
27. P. ENFLO, 'On the invariant subspace problem in Banach spaces', Acta Math. 158 (1987) 213-313.
28. L. A. FIALKOW, 'A note on the operator X → AX - XB', Trans. Amer. Math. Soc. 243 (1978) 147-168.
29. L. A. FIALKOW, 'A note on norm ideals and the operator X → AX - XB', Israel J. Math. 32 (1979) 331-348.
30. L. A. FIALKOW, 'Elements of spectral theory for generalized derivations', J. Operator Theory 3 (1980) 89-113.
31. L. A. FIALKOW, 'A note on the range of the operator X → AX - XB', Illinois J. Math. 25 (1981) 112-124.
32. H. FLANDERS and H. K. WIMMER, 'On the matrix equations AX - XB = C and AX - YB = C*', SIAM J. Appl. Math. 32 (1977) 707-710.
33. J. M. FREEMAN, 'The tensor product of semigroups and the operator equation SX - XT = A', J. Math. Mech. 19 (1970) 819-828.
34. W. X. GAO, 'Continued fraction solution of matrix equation AX - XB = C', Sci. China Ser. A 32 (1989) 1025-1035.
35. J. A. GOLDSTEIN, 'On the operator equation AX + XB = Q', Proc. Amer. Math. Soc. 70 (1978) 31-34.
36. G. H. GOLUB, S. NASH and C. F. VAN LOAN, 'A Hessenberg-Schur method for the problem AX + XB = C', IEEE Trans. Automat. Control 24 (1979) 909-913.
37. G. H. GOLUB and C. F. VAN LOAN, Matrix computations (2nd edn, Johns Hopkins University Press, Baltimore, MD, 1989).
38. S. GRABINER, 'Spectral consequences of the existence of intertwining operators', Comment. Math. Prace Mat. 22 (1980/81) 227-238.
39. W. HAHN, Stability of motion (Springer, New York, 1967).
40. P. R. HALMOS, A Hilbert space problem book (2nd edn, Springer, New York, 1982).
41. R. E. HARTE, 'Commutativity and separation of spectra II', Proc. Roy. Irish Acad. Sect. A 74 (1974) 239-244.
42. R. E. HARTWIG, 'Resultants and the solution of AX + XB = - C', SIAM J. Appl. Math. 23 (1972) 104-117.
43. R. E. HARTWIG, 'AX + XB = C, resultants and generalized inverses', SIAM J. Appl. Math. 28 (1975) 154-183.
44. M. L. J. HAUTUS, 'Operator substitution', Linear Algebra Appl. 205/206 (1994) 713-739.
45. J. Z. HEARON, 'Nonsingular solutions of TA - BT = C', Linear Algebra Appl. 16 (1977) 57-63.
46. E. HEINZ, 'Beiträge zur Störungstheorie der Spektralzerlegung', Math. Ann. 123 (1951) 415-438.
47. D. A. HERRERO, Approximation of Hilbert space operators, Vol. I, Pitman Res. Notes Math. Ser. 72 (Longman Sci. Tech., Harlow, 1982).
48. N. J. HIGHAM, 'Perturbation theory and backward error for AX - XB = C', BIT 33 (1993) 124-136.
49. N. J. HIGHAM, 'The matrix sign decomposition and its relation to the polar decomposition', Linear Algebra Appl. 212/213 (1994) 3-20.
50. T. B. HOOVER, 'Hyperinvariant subspaces for n-normal operators', Acta Sci. Math. (Szeged) 32 (1971) 109-119.
51. R. A. HORN and C. R. JOHNSON, Matrix analysis (Cambridge University Press, 1985).
52. D. Y. HU and L. REICHEL, 'Krylov-subspace methods for the Sylvester equation', Linear Algebra Appl. 172 (1992) 283-313.
53. J. JONES JR, 'Explicit solutions of the matrix equation AX + XB = C', Rend. Circ. Mat. Palermo (2) 23 (1974) 245-257.
54. -1 estimators', SIAM J. Matrix Anal. Appl. 13 (1992) 90-101.
55. M. G. KREIN, 'Some new studies of perturbation theory of self-adjoint operators', First Mathematical Summer School, Kiev, 1964 (Russian).
56. J. KYLE, 'Ranges of Lyapunov transformations for operator algebras', Glasgow Math. J 19 (1978) 129-134.
57. P. LANCASTER, Theory of matrices (Academic Press, New York, 1969).
58. P. LANCASTER, 'Explicit solutions of linear matrix equations', SIAM Rev. 12 (1970) 544-566.
59. P. LANCASTER, L. LERER and M. TISMENETSKY, 'Factored forms for solutions of AX - XB = C and X - AXB = C in companion matrices', Linear Algebra Appl. 62 (1984) 19-49.
60. J. P. LASALLE and S. LEFSHETZ, Stability by Lyapunov's direct method with applications (Academic Press, New York, 1961).
61. L. LERER and L. RODMAN, 'Sylvester and Lyapunov equations and some interpolation problems for rational matrix functions', Linear Algebra Appl. 185 (1993) 83-117.
62. R.-C. LI, 'New perturbation bounds for the unitary polar factor', SIAM J. Matrix Anal. Appl. 16 (1995) 327-332.
63. S. C. LIN and S. Y. SHAW, 'On the equations AX = Q and SX - XT = Q', J. Funct. Anal. 77 (1988) 352-363.
64. V. LOMONOSOV, 'Invariant subspaces for operators commuting with compact operators', Funktsional Anal. i Prilozhen 7 (1973) 55-56 (Russian); Functional Anal. Appl. 7 (1973) 213-214 (English).
65. V. LOMONOSOV, 'Invariant subspaces for operators commuting with compact operators', Funktsional Anal. i Prilozhen 7 (1973) 55-56 (Russian); Functional Anal. Appl. 7 (1973) 213-214 (English).
66. V. LOVASS-NAGY and D. L. POWERS, 'On least squares solutions of an inconsistent singular equation AX + XB = C', SIAM J. Appl. Math. 31 (1976) 84-88.
67. T. X. LU, 'Solution of the matrix equation AX - XB = C', Computing 37 (1986) 351-355.
68. G. LUMER and M. ROSENBLUM, 'Linear operator equations', Proc. Amer. Math. Soc. 10 (1959) 32-41.
69. A. LYAPUNOV, Problemes général de la stabilité du mouvement (1892); reprinted as Ann. of Math. Stud. 17 (Princeton University Press, 1947).
70. A. LYAPUNOV, Problemes général de la stabilité du mouvement (1892); reprinted as Ann. of Math. Stud. 17 (Princeton University Press, 1947).
71. E. MA, 'A finite series solution of the matrix equation AX - XB = C', SIAM J. Appl. Math. 14 (1966) 490-495.
72. C. C. MACDUFFEE, The theory of matrices (Springer, Berlin, 1933).
73. T. MAZUMDAR, 'Existence of solutions to the operator equation AX - XB = Q with possibly unbounded A, B, Q', Comput. Math. Appl. 19 (1990) 105-116.
74. A. OSTROWSKI and H. SCHNEIDER, 'Some theorems on the inertia of general matrices', J. Math. Anal. Appl. 4 (1962) 72-84.
75. W. V. PARKER, 'The matrix equation AX = XB', Duke Math. J. 17 (1950) 43-51.
76. G. K. PEDERSEN, 'On the operator equation HT + TH = 2K', Indiana Univ. Math. J. 25 (1976) 1029-1033.
77. V. Q. PHÓNG, 'The operator equation AX - XB = C with unbounded operators A and B and related abstract Cauchy problems', Math. Z. 208 (1991) 567-588.
78. D. L. POWERS, 'Solving AX + XB = C by control and tearing', Internat. J. Control 23 (1976) 421-425.
79. H. RADJAVI and P. ROSENTHAL, 'Hyperinvariant subspaces for spectral and n-normal operators', Acta Sci. Math. (Szeged) 32 (1971) 121-126.
80. H. RADJAVI and P. ROSENTHAL, Invariant subspaces (Springer, Heidelberg, 1973).
81. C. J. READ, 'A solution to the invariant subspace problem', Bull. London Math. Soc. 16 (1984) 335-348.
82. J. D. ROBERTS, 'Linear model reduction and solution of the algebraic Riccati equation by use of the sign function', Internat. J. Control 32 (1980) 677-687. (First issued as a Technical Report, Department of Engineering, University of Cambridge, 1971.)
83. M. ROSENBLUM, 'On the operator equation BX - XA = Q', Duke Math. J. 23 (1956) 263-270.
84. M. ROSENBLUM, 'The operator equation BX - XA = Q with self-adjoint A and B', Proc. Amer. Math. Soc. 20 (1969) 115-120.
85. P. ROSENTHAL, 'On the equations X = KXS and AX = XK', Proc. Banach Centre, Symposium on Spectral Theory, Warsaw (1982) 389-391.
86. W. E. ROTH, 'The equations AX - YB = C and AX - XB = C in matrices', Proc. Amer. Math. Soc. 3 (1952) 392-396.
87. W. RUDIN, Real and complex analysis (3rd edn, McGraw-Hill, New York, 1991).
88. D. E. RUTHERFORD, 'On the solution of the matrix equation AX + XB = C', Nederl. Akad. Wetensch. Proc. Ser. A 35 (1932) 53-59.
89. A. SCHWEINSBERG, 'The operator equation AX - XB = C with normal A and B', Pacific J. Math. 102 (1982) 447-453.
90. S. K. SEN and G. W. HOWELL, 'Direct fail-proof triangularization algorithms for AX + XB = C with error-free and parallel implementations', Appl. Math. Comput. 50 (1992) 255-278.
91. V. E. SHESTOPAL, 'Solution of the matrix equation AX - XB = C', Mat. Zametki 19 (1976) 449-451 (Russian).
92. S. SLAVOVA, G. BORISOVA and Z. ZELEV, 'On the operator equation AX = XB', Complex analysis and applications '87, Varna (1987) 474-476.
93. R. A. SMITH, 'Matrix equation XA + BX = C', SIAM J. Appl. Math. 16 (1968) 198-201.
94. E. D. SONTAG, Mathematical control theory (Springer, Berlin, 1990).
95. G. STARKE and W. NIETHAMMER, 'SOR for AX - XB = C', Linear Algebra Appl. 154/156 (1991) 355-375.
96. G. W. STEWART, 'Error and perturbation bounds for subspaces associated with certain eigenvalue problems', SIAM Rev. 15 (1973) 727-764.
97. G. W. STEWART and J.-G. SUN, Matrix perturbation theory (Academic Press, New York, 1990).
98. J. SYLVESTER, 'Sur l'equation en matrices px = xq', C. R. Acad. Sci. Paris 99 (1884) 67-71, 115-116.
99. B. SZ.-NAGY, 'Über die Ungleichung von H. Bohr', Math. Nachr. 9 (1953) 225-259.
100. J. D. VAALER, 'Some extremal functions in Fourier analysis', Bull. Amer. Math. Soc. 12 (1985) 183-216.
101. J. M. VARAH, 'On the separation of two matrices', SIAM J. Numer. Anal. 16 (1979) 216-222.
102. J. H. M. WEDDERBURN, 'Note on the linear matrix equation', Proc. Edinburgh Math. Soc. 22 (1904) 49-53.
103. J. P. WILLIAMS, 'Similarity and the numerical range', J. Math. Anal. Appl. 26 (1969) 307-314.
104. W. F. WONHAM, Linear multivariable control - a geometric approach (Springer, Berlin, 1979).