Abstract
We investigate the computational complexity of the minimal polynomial of an integer matrix.
We show that the computation of the minimal polynomial is in AC 0(GapL), the AC 0-closure of the logspace counting class GapL, which is contained in NC 2. Our main result is that the problem is hard for GapL (under AC 0 many-one reductions). The result extends to the verification of all invariant factors of an integer matrix.
Furthermore, we consider the complexity to check whether an integer matrix is diagonalizable. We show that this problem lies in AC 0(GapL) and is hard for AC 0(C = L) (under AC 0 many-one reductions).
This work was supported by the Deutsche Forschungsgemeinschaft
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
E. Allender, R. Beals, and M. Ogihara. The complexity of matrix rank and feasible systems of linear equations. Computational Complexity, 8:99–126, 1999.
C. Byrnes and M. Gauger. Characteristic free, improved decidability criteria for the similarity problem. Linear and Multilinear Algebra, 5:153–158, 1977.
R. Brualdi and H. Ryser. Combinatorial Matrix Theory, volume 39 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1991.
D. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs, Theory and Application. Academic Press, 1980.
F. Gantmacher. The Theory of Matrices, volume 1 and 2. AMS Chelsea Publishing, 1977.
A. Graham. Kronnecker Products and Matrix Calculus With Applications. Ellis Horwood Ltd., 1981.
T. M. Hoang and T. Thierauf. The complexity of verifying the characteristic polynomial and testing similarity. In 15th IEEE Conference on Computational Complexity (CCC), pages 87–95. IEEE Computer Society Press, 2000.
E. Kaltofen and B. Saunders. Fast parallel computation of hermite and smith forms of polynomial matrices. SIAM Algebraic and Discrete Methods, 8:683–690, 1987.
M. Santha and S. Tan. Verifying the determinant in parallel. Computational Complexity, 7:128–151, 1998.
A. Storjohann. An O(n 3) algorithm for frobenius normal form. In International Symposium on Symbolic and Algebraic Computation (ISSAC), 1998.
S. Toda. Counting problems computationally equivalent to the determinant. Technical Report CSIM 91-07, Dept. of Computer Science and Information Mathematics, University of Electro-Communications, Chofu-shi, Tokyo 182, Japan, 1991.
L. Valiant. Why is boolean complexity theory difficult. In M. S. Paterson, editor, Boolean Function Complexity, London Mathematical Society Lecture Notes Series 169. Cambridge University Press, 1992.
G. Villard. Fast parallel algorithms for matrix reduction to normal forms. Applicable Algebra in Engineering Communication and Computing (AAECC), 8:511–537, 1997.
J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hoang, T.M., Thierauf, T. (2001). The Complexity of the Minimal Polynomial. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_36
Download citation
DOI: https://doi.org/10.1007/3-540-44683-4_36
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42496-3
Online ISBN: 978-3-540-44683-5
eBook Packages: Springer Book Archive