On product of companion matrices

Lim, Arthur; Dai, Jialing

doi:10.1016/j.laa.2011.05.021

Cited by 11 publications

(12 citation statements)

References 5 publications

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“…All extensions involve powers of companion matrices. The formula derived in [11] can be used to calculate entries of an integer power of a companion matrix directly.…”

Section: Applicationsmentioning

confidence: 99%

Companion matrices and their relations to Toeplitz and Hankel matrices

Luo

Hill

2015

Special Matrices

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Abstract:In this paper we describe some properties of companion matrices and demonstrate some special patterns that arise when a Toeplitz or a Hankel matrix is multiplied by a related companion matrix. We present a necessary and sufficient condition, generalizing known results, for a matrix to be the transforming matrix for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel structure respectively.

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“…All extensions involve powers of companion matrices. The formula derived in [11] can be used to calculate entries of an integer power of a companion matrix directly.…”

Section: Applicationsmentioning

confidence: 99%

Companion matrices and their relations to Toeplitz and Hankel matrices

Luo

Hill

2015

Special Matrices

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“…One can use combinatorial arguments to derive combinatorial formulas for the entries of any power of C (see [2] and [6]). Although such combinatorial formulas are very interesting, they don't lead to fast algorithms to compute powers of C.…”

Section: Computing C Nmentioning

confidence: 99%

A fast elementary algorithm for computing the determinant of Toeplitz matrices

Cinkir

2014

Journal of Computational and Applied Mathematics

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Abstract. In recent years, a number of fast algorithms for computing the determinant of a Toeplitz matrix were developed. The fastest algorithm we know so far is of order k 2 log n + k 3 , where n is the number of rows of the Toeplitz matrix and k is the bandwidth size. This is possible because such a determinant can be expressed as the determinant of certain parts of n-th power of a related k × k companion matrix. In this paper, we give a new elementary proof of this fact, and provide various examples. We give symbolic formulas for the determinants of Toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when k is small.

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“…In the last years, the product of companion matrices has attracted much attention, because this product occurs in various fields of mathematics and applied sciences, such that the Floquet system theory related to the linear difference equations (see [4,5,16,17]). Diverse methods for computing the product of companion matrices have been proposed in the literature.…”

Section: Introductionmentioning

confidence: 99%

“…Diverse methods for computing the product of companion matrices have been proposed in the literature. For instance, in [16], the authors developed an explicit formula for the entries of the product of companion matrices. Then, they applied their results to solve linear difference equations of variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

Periodic matrix difference equations and companion matrices in blocks: some applications

Benkhaldoun¹,

Taher²,

Rachidi³

2021

Arab. J. Math.

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This study is devoted to some periodic matrix difference equations, through their associated product of companion matrices in blocks. Linear recursive sequences in the algebra of square matrices in blocks and the generalized Cayley–Hamilton theorem are considered for working out some results about the powers of matrices in blocks. Two algorithms for computing the finite product of periodic companion matrices in blocks are built. Illustrative examples and applications are considered to demonstrate the effectiveness of our approach.

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On product of companion matrices

Cited by 11 publications

References 5 publications

Companion matrices and their relations to Toeplitz and Hankel matrices

Companion matrices and their relations to Toeplitz and Hankel matrices

A fast elementary algorithm for computing the determinant of Toeplitz matrices

Periodic matrix difference equations and companion matrices in blocks: some applications

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