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Stability of Invariant Subspaces of Quaternion Matrices

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson W.N. Jr, Duffin R.J.: Series and parallel addition of matrices. J. Math. Anal. Appl. 26, 576–594 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bart H., Gohberg I., Kaashoek M.A.: Minimal Factorization of Matrix and Operator Functions, OT 1. Birkhäuser, Boston (1979)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Brenner J.L.: Matrices of quaternions. Pac. J. Math. 1, 329–335 (1951)

    MATH  Google Scholar 

  8. Campbell S., Daughtry J.: The stable solutions of quadratic matrix equations. Proc. Am. Math. Soc. 74(1), 19–23 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Conway J.B.: Finite-dimensional points of continuity of Gen and Com. Linear Algebra Appl. 35, 121–127 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Conway J.B., Halmos P.R.: Finite-dimensional points of continuity of Lat. Linear Algebra Appl. 31, 93–102 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Farenick D.R., Pidkowich B.A.F.: The spectral theorem in quaternions. Linear Algebra Appl. 371, 75–102 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Freiling G.: A survey of nonsymmetric Riccati equations. Linear Algebra Appl. 351/352, 243–270 (2002)

    Article  MathSciNet  Google Scholar 

  14. Gohberg I., Lancaster P., Rodman L.: Spectral analysis of matrix polynomials, I. Canonical forms and divisors. Linear Algebra Appl. 20, 1–44 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, 1982; republication, SIAM (2009)

  16. Gohberg, I., Lancaster, P., Rodman, L.: Invariant Subspaces of Matrices with Applications. Wiley, 1986; republication, SIAM (2006)

  17. Gohberg I., Rodman L.: On the distance between lattices of invariant subspaces of matrices. Linear Algebra Appl. 76, 85–120 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gohberg I., Rubinstein S.: Stability of minimal fractional decompositions of rational matrix functions. Oper. Theory: Adv. Appl. 18, 249–270 (1985)

    MathSciNet  Google Scholar 

  19. Gracia J.-M., Velasco F.E.: Stability of controlled invariant subspaces. Linear Algebra Appl. 418(2–3), 416–434 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hungerford T.W.: Algebra. Springer, New York (1974)

    MATH  Google Scholar 

  21. Johnson R.E.: On the equation χα = γ χ + β over an algebraic division ring. Bull Am. Math. Soc. 50, 202–207 (1944)

    Article  MATH  Google Scholar 

  22. Juang J., Lin W.W.: Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl. 20(1), 228–243 (1999)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  24. Košir T., Plestenjak B.: On stability of invariant subspaces of commuting matrices. Linear Algebra Appl. 342, 133–147 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Noakes L., Chung K.Y.: Invariant subspaces: continuous stability implies smooth stability. Proc. Am. Math. Soc. 120(1), 119–126 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pereira R., Rocha P., Vettori P.: Algebraic tools for the study of quaternionic behavioral systems. Linear Algebra Appl. 400, 121–140 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pereira R., Vettori P.: Stability of quaternionic linear systems. IEEE Trans. Automat. Control 51(3), 518–523 (2006)

    Article  MathSciNet  Google Scholar 

  30. Piziak R., Odell P.L., Hahn R.: Constructing projections for sums and intersections. Comput. Math. Appl. 17, 67–74 (1999)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ran A.C.M., Rodman L.: Stability of invariant Lagrangian subspaces I. Oper. Theory: Adv. Appl. 32, 181–218 (1988)

    MathSciNet  Google Scholar 

  33. Ran A.C.M., Rodman L.: A class of robustness problems in matrix analysis. Oper. Theory: Adv. Appl. 134, 337–383 (2002)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  35. Ran A.C.M., Rodman L.: Stability of invariant Lagrangian subspaces. Oper. Theory: Adv. Appl. 40, 391–425 (1989)

    MathSciNet  Google Scholar 

  36. Ran A.C.M., Rodman L.: The rate of convergence of real invariant subspaces. Linear Algebra Appl. 207, 197–224 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ran A.C.M., Roozemond L.: On strong α-stability of invariant subspaces of matrices. Oper. Theory: Adv. Appl. 40, 427–435 (1989)

    MathSciNet  Google Scholar 

  39. Rodman L.: Stable invariant subspaces modulo a subspace. Oper. Theory: Adv. Appl. 19, 399–413 (1986)

    MathSciNet  Google Scholar 

  40. Stewart G.W.: On the continuity of the generalized inverse. SIAM J. Appl. Math. 17, 33–45 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wan Z.-X.: Geometry of Matrices. World Scientific, River Edge (1996)

    Book  MATH  Google Scholar 

  42. Wiegmann N.A.: Some theorems on matrices with real quaternion entries. Can. J. Math. 7, 191–201 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Leiba Rodman

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  1. Leiba Rodman
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Correspondence to Leiba Rodman.

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Communicated by Daniel Aron Alpay.

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Rodman, L. Stability of Invariant Subspaces of Quaternion Matrices. Complex Anal. Oper. Theory 6, 1069–1119 (2012). https://doi.org/10.1007/s11785-012-0233-y

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