The combinatorial power of the companion matrix

Using combinatorial methods, we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to Waring's formula on symmetric functions, the general solution to homogeneous lin...

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Bibliographic Details
Published in:Linear algebra and its applications 1996, Vol.232, p.261-278
Main Authors: Chen, William Y.C., Louck, James D.
Format: Article
Language:English
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Summary:Using combinatorial methods, we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to Waring's formula on symmetric functions, the general solution to homogeneous linear recurrence relations, the multiplicative inverse of formal power series, the generating function of compositions (of numbers), a unified approach to Chebyshev polynomials including two recently discovered classes that satisfy analogous smallest-norm and orthogonality properties subject to different weight functions, Dickson polynomials of various kinds arising from the theory of finite fields, combinatorial expansions of Toeplitz matrices, and the recent notion of cycle dissections involving a bijective study of Waring's formula.
ISSN:0024-3795
1873-1856
DOI:10.1016/0024-3795(95)90163-9