On the matrix equation f(X)=A

Evard, Jean-Claude; Uhlig, F.

doi:10.1016/0024-3795(92)90390-v

Cited by 17 publications

(13 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Any matrix X such that X n = A is called an nth root of A and the problem of determining all the nth roots X of a given matrix has been dealt with by many mathematicians ( [2,3], [4, pp. 120-122], [5], [6, pp. 231-239], [7], [8, pp.…”

Section: Introductionmentioning

confidence: 99%

Extraction of nth roots of 2×2 matrices

Choudhry

2004

Linear Algebra and its Applications

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Extraction of nth roots of 2×2 matrices

Choudhry

2004

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…Remarkable examples of (1) are the matrix equations X k = A, e X = A, and Xe X = A, which define the matrix kth root [22,16], the matrix logarithm [1], and the matrix Lambert W function [8], respectively. Existence and finiteness of real and complex solutions to (1) are discussed, along with other properties of this matrix equation, in the excellent treatise by Evard and Uhlig [7]. In order to better understand the computational properties of the matrices that satisfy (1), it is useful to distinguish the solutions that can be written as a polynomial of A, or primary solutions, from those that cannot, called nonprimary.…”

Section: Introductionmentioning

confidence: 99%

“…In order to better understand the computational properties of the matrices that satisfy (1), it is useful to distinguish the solutions that can be written as a polynomial of A, or primary solutions, from those that cannot, called nonprimary. A useful characterization of primary solutions in terms of their eigenvalues is provided in [7].…”

Section: Introductionmentioning

confidence: 99%

Computing primary solutions of equations involving primary matrix functions

Fasi

Iannazzo

2019

Linear Algebra and its Applications

View full text Add to dashboard Cite

The matrix equation f (X) = A, where f is an analytic function and A is a square matrix, is considered. Some results on the classification of solutions are provided. When f is rational, a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of A. For real data, the algorithm yields the real solutions using only real arithmetic. Numerical experiments show that the algorithm performs in a stable fashion when run in finite precision arithmetic.

show abstract

“…1]). The latter class has been extensively studied in the literature, and a number of theoretical results, such as existence, uniqueness, and a classification of real and complex solutions, are discussed by Evard and Uhlig [13]. To the best of our knowledge, no general algorithm for dealing with the equation f (X) = A exists, even though algorithms tailored to most special cases of interest are available.…”

mentioning

confidence: 99%

Substitution algorithms for rational matrix equations

Fasi¹,

Iannazzo²

2020

etna

View full text Add to dashboard Cite

We study equations of the form r(X) = A, where r is a rational function and A and X are square matrices of the same size. We develop two techniques for solving these equations by inverting (through a substitution strategy) two schemes for the evaluation of rational functions of matrices. For triangular matrices, the new methods yield the same computational cost as the evaluation schemes from which they are obtained. A general equation can be reduced to upper triangular form by exploiting the Schur decomposition of the given matrix. For real data, the algorithms rely on the real Schur decomposition in order to compute real solutions using only real arithmetic. Numerical experiments show that our implementations are faster than existing alternatives without sacrificing accuracy.

show abstract

On the matrix equation f(X)=A

Cited by 17 publications

References 20 publications

Extraction of nth roots of 2×2 matrices

Extraction of nth roots of 2×2 matrices

Computing primary solutions of equations involving primary matrix functions

Substitution algorithms for rational matrix equations

Contact Info

Product

Resources

About