Extraction of mth roots in matrix rings over fields

Otero, Daniel

doi:10.1016/0024-3795(90)90279-l

Cited by 10 publications

(10 citation statements)

References 9 publications

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“…231-239], [7], [8, pp. 94-97], [9]). However, explicit formulae giving X in terms of A and its elements are not generally known.…”

Section: Introductionmentioning

confidence: 99%

Extraction of nth roots of 2×2 matrices

Choudhry

2004

Linear Algebra and its Applications

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This paper is concerned with the determination of algebraic formulae giving all the solutions of the matrix equation X n = A where n is a positive integer greater than 2 and A is a 2 × 2 matrix with real or complex elements. If A is a 2 × 2 scalar matrix, the equation X n = A has infinitely many solutions and we obtain explicit formulae giving all the solutions. If A is a non-scalar 2 × 2 matrix, the equation X n = A has a finite number of solutions and we give a formula expressing all solutions in terms of A and the roots of a suitably defined nth degree polynomial in a single variable. This leads to very simple formulae for all the solutions when A is either a singular matrix or a non-singular matrix with two coincident eigenvalues. Similarly when n = 3 or 4, we get explicit algebraic formulae for all the solutions. We also determine the precise number of solutions in various cases.

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“…231-239], [7], [8, pp. 94-97], [9]). However, explicit formulae giving X in terms of A and its elements are not generally known.…”

Section: Introductionmentioning

confidence: 99%

Extraction of nth roots of 2×2 matrices

Choudhry

2004

Linear Algebra and its Applications

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“…From a rectangular structure R, using the bijection d from equation (28) above and denoting by R(s; t) the unique rectangle on the pair (s; t) guaranteed by (25), deÿne…”

Section: Deriving An Algebramentioning

confidence: 99%

“…Special cases arise from a ne planes and k-nets described in [3], the matrices from Shader's work [36], and various classes of matrices considered by people constructing special solutions to Ho man's matrix question, see for instance [16,17,20,28,34,41,42] for general results.…”

Section: Generating Rectangular Structuresmentioning

confidence: 99%

Efficient exhaustive listings of reversible one dimensional cellular automata

Boykett

2004

Theoretical Computer Science

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This paper looks at an algebraic formulation of one dimensional cellular automata. Using the formulation connections to combinatorial structures and graph theory become clear. Strong results about uniqueness and isomorphism allows us to outline e ective algorithms for the generation of exhaustive lists of reversible one dimensional cellular automata, and to count the number of distinct examples that exist. These algorithms use the "orderly algorithm" methods to avoid the pitfalls of brute force searches.

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“…The existence of solutions x of the power equations of the type x m = A for the given matrix A, i.e. the existence of roots of matrices of degree m, was considered in [3], [15] and [16]. In [15] and [16] the authors discuss the solvability of power equations in the groups GL(n, K) in terms of the characteristic polynomial of matrix A.…”

Section: Introductionmentioning

confidence: 99%

“…the existence of roots of matrices of degree m, was considered in [3], [15] and [16]. In [15] and [16] the authors discuss the solvability of power equations in the groups GL(n, K) in terms of the characteristic polynomial of matrix A. In [3] the necessary and sufficient conditions for solvability of power and commutator equations in groups UT(n, K) are provided.…”

Section: Introductionmentioning

confidence: 99%