Consistency and efficient solution of the Sylvester equation for {star operator}-congruence

Electronic Journal of Linear Algebra

Volumn 22, Issue , 2011, Pages 849-863

  de Terán, Fernando ^a   Dopico, Froilán M ^a  

^a UNIVERSIDAD CARLOS III DE MADRID   (Spain)

Author keywords

Congruence of matrices; Generalized Schur decomposition; Matrix equations; Palindromic eigenvalue problems; Sylvester equation

Indexed keywords

EID: 80053164079     PISSN: None     EISSN: 10813810     Source Type: Journal    
DOI: None     Document Type: Review

References (29)

1. C.S. Ballantine. A note on the matrix equation H = AP +PA*. Linear Algebra Appl., 2:37-47, 1969.
2. R.H. Bartels and G.W. Stewart. Algorithm 432: Solution of the matrix equation AX+XB = C. Comm. ACM, 15:820-826, 1972.
3. M.A. Beitia and J.-M. Gracia. Sylvester matrix equation for matrix pencils. Linear Algebra Appl., 232:155-197, 1996.
4. TA = B. SIAM J. Matrix Anal. Appl., 20:295-302, 1999.
5. R. Byers and D. Kressner. Structured condition numbers for invariant subspaces. SIAM J. Matrix Anal. Appl., 28:326-347, 2006.
6. C.-Y. Chiang, E.K.-W. Chu, and W.-W. Lin. On the {star operator}-Sylvester equation AX + X*B{star operator} = C. Submitted, 2011. Available at http://oz.nthu.edu.tw/~d947207/colloquium/20081020/ TSylvester_081013.pdf.
7. D.S. Cvetković-Ilić. The solutions of some operator equations. J. Korean Math. Soc., 45:1417-1425, 2008.
8. J.W. Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997.
9. T = 0 and its application to the theory of orbits. Linear Algebra Appl., 434:44-67, 2011.
10. F. De Terán and F.M. Dopico. The equation XA+AX*= 0 and the dimension of *-congruence orbits. Electron. J. Linear Algebra, 22:448-465, 2011.
11. D.S. Djordjević. Explicit solution of the operator equation A*X+X*A = B. J. Comput. Appl. Math., 200:701-704, 2007.
12. H. Flanders and H.K. Wimmer. On the matrix equations AX-XB = C and AX-Y B = C. SIAM J. Appl. Math., 32:707-710, 1977.
13. I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.
14. G.H. Golub, S. Nash, and C.F. Van Loan. A Hessenberg-Schur method for the problem AX + XB = C. IEEE Trans. Automat. Control, 24:909-913, 1979.
15. G.H. Golub and C.F. Van Loan. Matrix Computations, Third Edition. Johns Hopkins University Press, Baltimore, MD, USA, 1996.
16. N.J. Higham. Perturbation theory and backward error for AX-XB = C. BIT, 33:124-136, 1993.
17. N.J. Higham. Accuracy and Stability of Numerical Algorithms, Second Edition. SIAM, Philadelphia, PA, 2002.
18. J.H. Hodges. Some matrix equations over a finite field. Ann. Mat. Pura Appl., 44:245-250, 1957.
19. R.A. Horn and C.R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991.
20. R.A. Horn and V.V. Sergeichuk. Canonical forms for complex matrix congruence and *congruence. Linear Algebra Appl., 416:1010-1032, 2006.
21. I. Jonsson and B. Kågström. Recursive blocked algorithms for solving triangular systems. Part I: One-sided and coupled Sylvester-type matrix equations. ACM Trans. Math. Software, 28:392-415, 2002.
22. B. Kågström and L. Westin. Generalized Schur methods with condition estimators for solving the generalized Sylvester equation. IEEE Trans. Automat. Control, 34:745-751, 1989.
23. D. Kressner, C. Schröder, and D.S. Watkins. Implicit QR algorithms for palindromic and even eigenvalue problems. Numer. Algorithms, 51:209-238, 2009.
24. W.E. Roth. The equations AX-Y B = C and AX-XB = C in matrices. Proc. Amer. Math. Soc., 3:392-396, 1952.
25. G.W. Stewart and J.-G Sun. Matrix Perturbation Theory. Academic Press, Inc., Boston. MA, 1990.
26. V.L. Syrmos and F.L. Lewis. Decomposability and quotient subspaces for the pencil sL-M. SIAM J. Matrix Anal. Appl., 15:865-880, 1994.
27. O. Taussky and H. Wielandt. On the matrix function AX+X′A′. Arch. Rational Mech. Anal., 9:93-96, 1962.
28. H.K. Wimmer. Consistency of a pair of generalized Sylvester equations. IEEE Trans. Automatic Control, 39:1014-1016, 1994.
29. H.K. Wimmer. Roth's theorems for matrix equations with symmetry constraints. Linear Algebra Appl., 199:357-362, 1994.