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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Au-Yeung, Y.H.: On the Eigenvalues and Numerical Range of a Quaternionic Matrix (1994) (preprint)
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)
Bell, P.: A Note on the Emptiness of Semigroup Intersections. Fundamenta Informaticae 79, 1–4 (2007)
Bell, P., Potapov, I.: On the Membership of Invertible Diagonal and Scalar Matrices. Theoretical Computer Science, 37–45 (2007)
Blondel, V., Megretski, A.: Unsolved problems in Mathematical Systems and Control Theory. Princeton University Press, Princeton, NJ (2004)
Blondel, V., Jeandel, E., Koiran, P., Portier, N.: Decidable and undecidable problems about quantum automata. SIAM Journal on Computing 34(6), 1464–1473 (2005)
Cassaigne, J., Harju, T., Karhumäki, J.: On the Undecidability of Freeness of Matrix Semigroups. Intern. J. Alg. & Comp. 9, 295–305 (1999)
D’Alessandro, F.: Free Groups of Quaternions. Intern. J. of Alg. and Comp. (IJAC) 14(1) (February 2004)
Halava, V., Harju, T.: On Markov’s Undecidability Theorem for Integer Matrices, TUCS Technical Report Number 758 (2006)
Halava, V., Harju, T., Hirvensalo, M.: Undecidability Bounds for Integer Matrices using Claus Instances, TUCS Technical Report 766 (2006)
Lengyel, E.: Mathematics for 3D Game Programming & Computer Graphics, Charles River Media (2004)
Markov, A.: On Certain Insoluble Problems Concerning Matrices. Doklady Akad. Nauk SSSR, 539–542 (1947)
Matiyasevich, Y., Senizergues, G.: Decision Problems for Semi-Thue Systems with a Few Rules. Theoretical Computer Science 330(1), 145–169 (2005)
Paterson, M.: Unsolvability in 3 × 3 Matrices. Studies in Applied Mathematics 49 (1970)
So, W., Thomson, R.C., Zhang, F.: Numerical Ranges of Matrices with Quaternion Entries. Linear and Multilinear Algebra 37, 175–195 (1994)
Swierczkowski, S.: A Class of Free Rotation Groups. Indag. Math. 5(2), 221–226 (1994)
Velichova, D., Zacharias, S.: Projection from 4D to 3D. Journal for Geometry and Graphics 4(1), 55–69 (2000)
Wiegmann, N.A.: Some Theorems on Matrices with Real Quaternion Elements. Can. Jour. Math. 7 (1955)
Zhang, F.: Quaternions and Matrices of Quaternions. Linear Algebra Appl. 251, 21–57 (1997)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bell, P., Potapov, I. (2007). Reachability Problems in Quaternion Matrix and Rotation Semigroups. In: Kučera, L., Kučera, A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74456-6_32
Download citation
DOI: https://doi.org/10.1007/978-3-540-74456-6_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74455-9
Online ISBN: 978-3-540-74456-6
eBook Packages: Computer ScienceComputer Science (R0)