Two remarks on matrix exponentials

Wermuth, Edgar

doi:10.1016/0024-3795(89)90554-5

Cited by 19 publications

(8 citation statements)

References 8 publications

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“…It may appear that with A and B as given on pg. 128 of [9], one has a counterexample to the above assertion. However, since both A and A + B do not satisfy the constraint −π < Im(λ i ) < −π , we have log(e A ) = A and log(e A+B ) = A + B.…”

Section: Preliminariesmentioning

confidence: 75%

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

Jog

2008

J Elasticity

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Section: Preliminariesmentioning

confidence: 75%

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

Jog

2008

J Elasticity

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“…When A and B commute, so must exp(A) and exp(B). Moreover, when A and B have algebraic entries, the converse also holds, as shown in (Wermuth 1989). Also, when A and B commute, it holds that exp(A) exp(B) = exp(A + B).…”

Section: Matrix Exponentialsmentioning

confidence: 86%

Solvability of Matrix-Exponential Equations

Ouaknine

Pouly

Sousa-Pinto

et al. 2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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We consider a continuous analogue of (Babai et al. 1996) 's and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, . . . , A k , C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals t1, . . . , t k such thatWe show that this problem is undecidable in general, but decidable under the assumption that the matrices A1, . . . , A k commute. Our results have applications to reachability problems for linear hybrid automata.Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the MinkowskiWeyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem.

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“…One of the drawbacks of those methods is that the usual functional equation 6®+' = e'e' which holds for complex scalars a and b is not true for matrices A and B in general unless we have that AB = BA holds. A detailed discussion of tliis can be found in [13]. Splitting methods make use of a fonnula that is due to Trotter [11].…”

Section: Introductionmentioning

confidence: 99%

A Splitting method for the calculation of the matrix exponential

Stichel¹

1994

Analysis

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The computation of the exponential of a m x m matrix A arises as an unportant problem in numerical analysis and system theory. There are a niunber of methods proposed in the literature for this ta.sk. Most of those methods suflEer from severe drawbacks. Under the more favourable methods there axe the SpUtting or Decomposition methods. It is tried to decompose the original matrix A into a sum of matrices A = Ai + ... + A/^ and to compute the exponential of each Ai separately. One of the drawbacks of this method is that the well-known functional equation of the exponential function does not hold for matrices in general. In this paper we will präsent a decomposition of A S C""*"* into a siim of matrices A = Ai + A2 such that e'^ = e^^e^^ holds. This decomposition is based on the sign fimction of a matrix. Moreover it may be used that the matrices A, in general are rank deficient. If m; < m is the rank of the matrix Ai we only have to compute the exponential of two m; x m; (i = 1,2) instead of two rank deficient m x m matrices. The method is especiaJly useful if e^' has to be computed for varioiis values of tec.

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Two remarks on matrix exponentials

Cited by 19 publications

References 8 publications

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

Solvability of Matrix-Exponential Equations

A Splitting method for the calculation of the matrix exponential

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