Complex Analysis and Operator Theory

, Volume 6, Issue 5, pp 1069–1119 | Cite as

Stability of Invariant Subspaces of Quaternion Matrices

  • Leiba RodmanEmail author


A quaternion invariant subspace of a quaternion matrix is said to be stable (in the sense of robustness) if every nearby matrix has an invariant subspace close to the original one. Under mild hypothesis, necessary and sufficient conditions are given for quaternion invariant subspaces to be stable. Other notions of stability of quaternion invariant subspaces are studied as well, and stability criteria developed. Applications to robustness of solutions of certain classes of quaternion matrix equations are given.


Quaternion matrix Invariant subspace Matrix equation 

Mathematics Subject Classification

15A24 15B33 15A99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alam R., Bora S., Karow M., Mehrmann V., Moro J.: Perturbation theory for Hamiltonian matrices and the distance to bounded-realness. SIAM J. Matrix Anal. Appl. 32, 484–514 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alpay D., Ran A.C.M., Rodman L.: Basic classes of matrices with respect to quaternionic indefinite inner product spaces. Linear Algebra Appl. 416, 242–269 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Anderson W.N. Jr, Duffin R.J.: Series and parallel addition of matrices. J. Math. Anal. Appl. 26, 576–594 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bart H., Gohberg I., Kaashoek M.A.: Minimal Factorization of Matrix and Operator Functions, OT 1. Birkhäuser, Boston (1979)Google Scholar
  5. 5.
    Bart H., Gohberg I., Kaashoek M.A.: Stable factorizations of monic matrix polynomials and stable invariant subspaces. Integr. Equ. Oper. Theory 1(4), 496–517 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bart H., Gohberg I., Kaashoek M.A., Ran A.C.M.: Factorization of Matrix and Operator Functions: The State Space Method, OT 178. Birkhäuser, Boston (2008)Google Scholar
  7. 7.
    Brenner J.L.: Matrices of quaternions. Pac. J. Math. 1, 329–335 (1951)zbMATHGoogle Scholar
  8. 8.
    Campbell S., Daughtry J.: The stable solutions of quadratic matrix equations. Proc. Am. Math. Soc. 74(1), 19–23 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Constales D.: A closed formula for the Moore-Penrose generalized inverse of a complex matrix of given rank. Acta Math. Hungar. 80(1–2), 83–88 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Conway J.B.: Finite-dimensional points of continuity of Gen and Com. Linear Algebra Appl. 35, 121–127 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Conway J.B., Halmos P.R.: Finite-dimensional points of continuity of Lat. Linear Algebra Appl. 31, 93–102 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Farenick D.R., Pidkowich B.A.F.: The spectral theorem in quaternions. Linear Algebra Appl. 371, 75–102 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Freiling G.: A survey of nonsymmetric Riccati equations. Linear Algebra Appl. 351/352, 243–270 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gohberg I., Lancaster P., Rodman L.: Spectral analysis of matrix polynomials, I. Canonical forms and divisors. Linear Algebra Appl. 20, 1–44 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, 1982; republication, SIAM (2009)Google Scholar
  16. 16.
    Gohberg, I., Lancaster, P., Rodman, L.: Invariant Subspaces of Matrices with Applications. Wiley, 1986; republication, SIAM (2006)Google Scholar
  17. 17.
    Gohberg I., Rodman L.: On the distance between lattices of invariant subspaces of matrices. Linear Algebra Appl. 76, 85–120 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Gohberg I., Rubinstein S.: Stability of minimal fractional decompositions of rational matrix functions. Oper. Theory: Adv. Appl. 18, 249–270 (1985)MathSciNetGoogle Scholar
  19. 19.
    Gracia J.-M., Velasco F.E.: Stability of controlled invariant subspaces. Linear Algebra Appl. 418(2–3), 416–434 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hungerford T.W.: Algebra. Springer, New York (1974)zbMATHGoogle Scholar
  21. 21.
    Johnson R.E.: On the equation χα = γ χ + β over an algebraic division ring. Bull Am. Math. Soc. 50, 202–207 (1944)zbMATHCrossRefGoogle Scholar
  22. 22.
    Juang J., Lin W.W.: Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl. 20(1), 228–243 (1999)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kaashoek M.A., van der Mee C.V.M., Rodman L.: Analytic operator functions with compact spectrum. II. Spectral pairs and factorization. Integr. Equ. Oper. Theory 5(6), 791–827 (1982)zbMATHCrossRefGoogle Scholar
  24. 24.
    Košir T., Plestenjak B.: On stability of invariant subspaces of commuting matrices. Linear Algebra Appl. 342, 133–147 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mehl C., Mehrmann V., Ran A.C.M., Rodman L.: Perturbation analysis of Lagrangian invariant subspaces of symplectic matrices. Linear Multilinear Algebra 57(2), 141–184 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Mehl C., Mehrmann V., Ran A.C.M., Rodman L.: Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations. Linear Algebra Appl. 435, 687–716 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Noakes L., Chung K.Y.: Invariant subspaces: continuous stability implies smooth stability. Proc. Am. Math. Soc. 120(1), 119–126 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Pereira R., Rocha P., Vettori P.: Algebraic tools for the study of quaternionic behavioral systems. Linear Algebra Appl. 400, 121–140 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Pereira R., Vettori P.: Stability of quaternionic linear systems. IEEE Trans. Automat. Control 51(3), 518–523 (2006)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Piziak R., Odell P.L., Hahn R.: Constructing projections for sums and intersections. Comput. Math. Appl. 17, 67–74 (1999)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ran A.C.M., Rodman L.: Rate of stability of solutions of matrix polynomial and quadratic equations. Integr. Equ. Oper. Theory 27(1), 71–102 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Ran A.C.M., Rodman L.: Stability of invariant Lagrangian subspaces I. Oper. Theory: Adv. Appl. 32, 181–218 (1988)MathSciNetGoogle Scholar
  33. 33.
    Ran A.C.M., Rodman L.: A class of robustness problems in matrix analysis. Oper. Theory: Adv. Appl. 134, 337–383 (2002)MathSciNetGoogle Scholar
  34. 34.
    Ran A.C.M., Rodman L.: On the index of conditional stability of stable invariant Lagrangian subspaces. SIAM J. Matrix Anal. Appl. 29(4), 1181–1190 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ran A.C.M., Rodman L.: Stability of invariant Lagrangian subspaces. Oper. Theory: Adv. Appl. 40, 391–425 (1989)MathSciNetGoogle Scholar
  36. 36.
    Ran A.C.M., Rodman L.: The rate of convergence of real invariant subspaces. Linear Algebra Appl. 207, 197–224 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Ran A.C.M., Rodman L., Rubin A.L.: Stability index of invariant subspaces of matrices. Linear Multilinear Algebra 36(1), 27–39 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Ran A.C.M., Roozemond L.: On strong α-stability of invariant subspaces of matrices. Oper. Theory: Adv. Appl. 40, 427–435 (1989)MathSciNetGoogle Scholar
  39. 39.
    Rodman L.: Stable invariant subspaces modulo a subspace. Oper. Theory: Adv. Appl. 19, 399–413 (1986)MathSciNetGoogle Scholar
  40. 40.
    Stewart G.W.: On the continuity of the generalized inverse. SIAM J. Appl. Math. 17, 33–45 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Wan Z.-X.: Geometry of Matrices. World Scientific, River Edge (1996)zbMATHCrossRefGoogle Scholar
  42. 42.
    Wiegmann N.A.: Some theorems on matrices with real quaternion entries. Can. J. Math. 7, 191–201 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Zhang F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsThe College of William and MaryWilliamsburgUSA

Personalised recommendations