Abstract
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.
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References
S. J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Information Processing Letters, 18:147–150, 1984.
S. Chaiken. A combinatorial proof of the all minors matrix theorem. SIAM J. Algebraic Discrete Methods, 3:319–329, 1982.
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.
L. Csanky. Fast parallel inversion algorithm. SIAM J of Computing, 5:818–823, 1976.
C. Damm. DET=L(#L). Technical Report Informatik-Preprint 8, Fachbereich Informatik der Humboldt-UniversitÄt zu Berlin, 1991.
D. Fadeev and V. Fadeeva. Computational Methods in Linear Algebra. Freeman, San Francisco, 1963.
D. Foata. Etude algébrique de certains problèmes d'analyse combinatoire et du calcul des probabilités. Publ. Inst. Statist. Univ. Paris, 14:81–241, 1965.
D. Foata. A combinatorial proof of Jacobi's identity. Ann. Discrete Math., 6:125–135, 1980.
A. Garsia and ö. Egecioglu. Combinatorial foundations of computer science. unpublished collection.
I. Gessel. Tournaments and Vandermonde's determinant. J Graph Theory, 3:305–307, 1979.
D. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, New York, 1992.
T. F. Leighton. Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann Publishers Inc., San Mateo, 1992.
M. Minoux. Bideterminants, arborescences and extension of the matrix-tree theorem to semirings. Discrete Mathematics, 171:191–200, 1997.
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.
J. B. Orlin. Line-digraphs, arborescences, and theorems of Tutte and Knuth. J. Combin. Theory Ser. B, 25:187–198, 1978.
P. A. Samuelson. A method of determining explicitly the coefficients of the characteristic polynomial. Ann. Math. Stat., 13:424–429, 1942.
V. Strassen. Vermeidung von divisionen. Journal of Reine U. Angew Math, 264:182–202, 1973.
H. Straubing. A combinatorial proof of the Cayley-Hamilton theorem. Discrete Maths., 43:273–279, 1983.
D. Stanton and D. White. Constructive Combinatorics. Springer-Verlag, 1986.
H. N. V. Tempereley. Graph Theory and Applications. Ellis Horwood, Chichester, 1981.
S. Toda. Counting problems computationally equivalent to the determinant. manuscript, 1991.
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.
V Vinay. Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In Proc. 6th Structure in Complexity Theory Conference, pages 270–284, 1991.
D. Zeilberger. A combinatorial approach to matrix algebra. Discrete Mathematics, 56:61–72, 1985.
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Mahajan, M., Vinay, V. (1998). Determinant: Old algorithms, new insights. In: Arnborg, S., Ivansson, L. (eds) Algorithm Theory — SWAT'98. SWAT 1998. Lecture Notes in Computer Science, vol 1432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054375
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DOI: https://doi.org/10.1007/BFb0054375
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