Determinant: Old algorithms, new insights
In this paper we approach the problem of computing the characteristic polynomial of a matrix from the combinatorial viewpoint. We present several combinatorial characterizations of the coefficients of the characteristic polynomial, in terms of walks and closed walks of different kinds in the underlying graph. We develop algorithms based on these characterizations, and show that they tally with well-known algorithms arrived at independently from considerations in linear algebra.
KeywordsCharacteristic Polynomial Tour Sequence Combinatorial Proof Combinatorial Interpretation Matrix Power
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- [Chi85]A. L. Chistov. Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic. In Proc Int. Conf. Foundations of Computation Theory, LNCS 199, pages 63–69. Springer, 1985.Google Scholar
- [Dam91]C. Damm. DET=L(#L). Technical Report Informatik-Preprint 8, Fachbereich Informatik der Humboldt-UniversitÄt zu Berlin, 1991.Google Scholar
- [FF63]D. Fadeev and V. Fadeeva. Computational Methods in Linear Algebra. Freeman, San Francisco, 1963.Google Scholar
- [GE]A. Garsia and ö. Egecioglu. Combinatorial foundations of computer science. unpublished collection.Google Scholar
- [Koz92]D. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1992.Google Scholar
- [MV97]M. Mahajan and V Vinay. Determinant: combinatorics, algorithms, complexity. Chicago Journal of Theoretical Computer Science http://cs-www.uchicago.edu/publications/cjtcs, 1997:5, 1997. A preliminary version appeared as “A combinatorial algorithm for the determinant” in Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms SODA97.MathSciNetGoogle Scholar
- [SW86]D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, 1986.Google Scholar
- [Tem81]H. N. V. Tempereley. Graph Theory and Applications. Ellis Horwood, Chichester, 1981.Google Scholar
- [Tod91]S. Toda. Counting problems computationally equivalent to the determinant. manuscript, 1991.Google Scholar
- [Val92]L. G. Valiant. Why is boolean complexity theory difficult? In M. S. Paterson, editor, Boolean Function Complexity. Cambridge University Press, 1992. London Mathematical Society Lecture Notes Series 169.Google Scholar
- [Vin91]V Vinay. Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In Proc. 6th Structure in Complexity Theory Conference, pages 270–284, 1991.Google Scholar