Subalgebras of matrix algebras generated by companion matrices

Guersenzvaig, Natalio H.; Szechtman, Fernando

doi:10.1016/j.laa.2009.12.019

Cited by 3 publications

(12 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Lemma 3.2 of [GS2], f divides p n−1 s and hence p n−1 g. It follows that h divides p n−1 , so p n−1 = 0. Proceeding like this with e 2 , ..., e n−1 we see that p n−2 = p n−3 = ... = p 1 = 0.…”

Section: Corollarymentioning

confidence: 89%

“…We let R C, D stand for the subalgebra of M n (R) generated by C and D, and R[C, D] for the R-span of all matrices C i D j , where 0 ≤ i, j < n. It is shown in Corollary 6.3 of [GS2] that R C, D = R [C, D]. Moreover, Theorem 9.2 of [GS2] shows that if R is a unique factorization domain then R C, D is necessarily a free R-module of rank n + (n − m)(n − 1), and Theorem 9.3 of [GS2] gives a presentation for R C, D in terms of C, D. As indicated in the Introduction, we wish to extend these results to the more general setting of integrally closed domains as well as furnishing a counterexample to the freeness of R C, D when R = Z[ √ 5].…”

Section: Corollarymentioning

confidence: 99%

“…, n − 1. It follows from Lemma 6.1 of [GS2] that S = R [A, B]. From f (A) = 0 and g(B) = 0 it follows that S is R-spanned by A i B j , 0 ≤ i, j ≤ n − 1.…”

Section: Proofmentioning

confidence: 99%

“…Given monic polynomials f, g of degree n over a commutative ring with identity R, let C and D stand for their respective companion matrices. Recently, [GS2] studied the subalgebra R C, D of M n (R) generated by C and D. It is shown in [GS2] that R C, D always coincides with R[C, D], the R-span of all matrices C i D j , where 0 ≤ i, j < n. Moreover, necessary and sufficient conditions for C, D to generate M n (R) were given, and a presentation for M n (R) in terms of C and D was furnished in that case. Furthermore, if R = Z and f, g are relatively prime then the sublattice Z C, D of M n (Z) was shown to have full rank n 2 and the exact value of the finite index [M n (Z) : Z C, D ] was found to be |R(f, g)| n−1 , where R(f, g) is the resultant to f and g.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Is every matrix similar to a polynomial in a companion matrix?

Guersenzvaig

Szechtman

2012

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

Given a field F , an integer n ≥ 1, and a matrix A ∈ M n (F ), are there polynomials f, g ∈ F [X], with f monic of degree n, such that A is similar to g(C f ), where C f is the companion matrix of f ? For infinite fields the answer is easily seen to positive, so we concentrate on finite fields. In this case we give an affirmative answer, provided |F | ≥ n−2. Moreover, for any finite field F , with |F | = m, we construct a matrix A ∈ M m+3 (F ) that is not similar to any matrix of the form g(C f ).Of use above, but also of independent interest, is a constructive procedure to determine the similarity type of any given matrix g(C f ) purely in terms of f and g, without resorting to polynomial roots in F or in any extension thereof. This, in turn, yields an algorithm that, given g and the invariant factors of any A, returns the elementary divisors of g(A). It is a rational procedure, as opposed to the classical method that uses the Jordan decomposition of A to find that of g(A).Finally, extending prior results by the authors, we show that for an integrally closed ring R with field of fractions F and companion matrices C, D the subalgebra R C, D of M n (R) is a free R-module of rank n + (n − m)(n − 1), where m is the degree of gcd(f, g) ∈ F [X], and a presentation for R C, D is given in terms of C and D. A counterexample is furnished to show that R C, D need not be a free R-module if R is not integrally closed. The preceding information is used to study M n (R), and others, as R[X]-modules.2000 Mathematics Subject Classification. 15A21, 15A72

show abstract

Section: Corollarymentioning

confidence: 89%

Section: Corollarymentioning

confidence: 99%

“…, n − 1. It follows from Lemma 6.1 of [GS2] that S = R [A, B]. From f (A) = 0 and g(B) = 0 it follows that S is R-spanned by A i B j , 0 ≤ i, j ≤ n − 1.…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Is every matrix similar to a polynomial in a companion matrix?

Guersenzvaig

Szechtman

2012

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, K(C) may be regarded as an algebra over F by the bilinearity of the mapping f : K(C) × K(C) → K(C) defined as f (K(C, a), K(C, b)) = K(C, b(C)a) (also see [8]). …”

Section: Krylov Matrices Of the Companion Matrixmentioning

confidence: 99%

A new aspect of Hankel matrices via Krylov matrix

Cheon

Kim

2013

Linear Algebra and its Applications

View full text Add to dashboard Cite

An investigation is made of the Krylov matrices associated to the companion matrix C of a monic polynomial p(x) of degree n. In this paper, we propose a new approach to the study of Hankel matrices as Krylov matrices of C T . Firstly we focus on algebraic structures of Krylov matrices of C over a field F. We then discuss an equivalent condition for the Hankel matrix to be a Krylov matrix of C T by the notion of compatibility. As a result, we derive new determinant formulas for such Hankel matrices in terms of eigenvalues and eigenvectors of C respectively.

show abstract

On the Splitting Ring of a Polynomial

Szechtman

2016

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

Abstract. Let f (Z) = Z n − a 1 Z n−1 + · · · + (−1) n−1 a n−1 Z + (−1) n an be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I f of R[X 1 , . . . , Xn] generated by σ i (X 1 , . . . , Xn) − a i , where σ 1 , . . . , σn are the elementary symmetric polynomials, as well as the quotient ring R[X 1 , . . . , Xn]/I f .

show abstract

Subalgebras of matrix algebras generated by companion matrices

Cited by 3 publications

References 9 publications

Is every matrix similar to a polynomial in a companion matrix?

Is every matrix similar to a polynomial in a companion matrix?

A new aspect of Hankel matrices via Krylov matrix

On the Splitting Ring of a Polynomial

Contact Info

Product

Resources

About