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Reachability Problems in Quaternion Matrix and Rotation Semigroups

  • Paul Bell
  • Igor Potapov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

We examine computational problems on quaternion matrix and rotation semigroups. It is shown that in the ultimate case of quaternion matrices, in which multiplication is still associative, most of the decision problems for matrix semigroups are undecidable in dimension two. The geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2 and 3-sphere. In particular, we show that the reachability of the rotation problem is undecidable on the 3-sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers.

Keywords

Unit Quaternion Matrix Problem Membership Problem Reachability Problem Quaternion Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Au-Yeung, Y.H.: On the Eigenvalues and Numerical Range of a Quaternionic Matrix (1994) (preprint)Google Scholar
  2. 2.
    Babai, L., Beals, R., Cai, J., Ivanyos, G., Luks, E.M.: Multiplicative Equations over Commuting Matrices. In: Proc. 7th ACM-SIAM Symp. on Discrete Algorithms, pp. 498–507. ACM, New York (1996)Google Scholar
  3. 3.
    Bell, P.: A Note on the Emptiness of Semigroup Intersections. Fundamenta Informaticae 79, 1–4 (2007)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bell, P., Potapov, I.: On the Membership of Invertible Diagonal and Scalar Matrices. Theoretical Computer Science, 37–45 (2007)Google Scholar
  5. 5.
    Blondel, V., Megretski, A.: Unsolved problems in Mathematical Systems and Control Theory. Princeton University Press, Princeton, NJ (2004)zbMATHGoogle Scholar
  6. 6.
    Blondel, V., Jeandel, E., Koiran, P., Portier, N.: Decidable and undecidable problems about quantum automata. SIAM Journal on Computing 34(6), 1464–1473 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cassaigne, J., Harju, T., Karhumäki, J.: On the Undecidability of Freeness of Matrix Semigroups. Intern. J. Alg. & Comp. 9, 295–305 (1999)zbMATHCrossRefGoogle Scholar
  8. 8.
    D’Alessandro, F.: Free Groups of Quaternions. Intern. J. of Alg. and Comp. (IJAC) 14(1) (February 2004)Google Scholar
  9. 9.
    Halava, V., Harju, T.: On Markov’s Undecidability Theorem for Integer Matrices, TUCS Technical Report Number 758 (2006)Google Scholar
  10. 10.
    Halava, V., Harju, T., Hirvensalo, M.: Undecidability Bounds for Integer Matrices using Claus Instances, TUCS Technical Report 766 (2006)Google Scholar
  11. 11.
    Lengyel, E.: Mathematics for 3D Game Programming & Computer Graphics, Charles River Media (2004)Google Scholar
  12. 12.
    Markov, A.: On Certain Insoluble Problems Concerning Matrices. Doklady Akad. Nauk SSSR, 539–542 (1947)Google Scholar
  13. 13.
    Matiyasevich, Y., Senizergues, G.: Decision Problems for Semi-Thue Systems with a Few Rules. Theoretical Computer Science 330(1), 145–169 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Paterson, M.: Unsolvability in 3 × 3 Matrices. Studies in Applied Mathematics 49 (1970)Google Scholar
  15. 15.
    So, W., Thomson, R.C., Zhang, F.: Numerical Ranges of Matrices with Quaternion Entries. Linear and Multilinear Algebra 37, 175–195 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Swierczkowski, S.: A Class of Free Rotation Groups. Indag. Math. 5(2), 221–226 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Velichova, D., Zacharias, S.: Projection from 4D to 3D. Journal for Geometry and Graphics 4(1), 55–69 (2000)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Wiegmann, N.A.: Some Theorems on Matrices with Real Quaternion Elements. Can. Jour. Math. 7 (1955)Google Scholar
  19. 19.
    Zhang, F.: Quaternions and Matrices of Quaternions. Linear Algebra Appl. 251, 21–57 (1997)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Paul Bell
    • 1
  • Igor Potapov
    • 1
  1. 1.Department of Computer Science, University of Liverpool, Ashton Building, Ashton St, Liverpool L69 3BXUK

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