A useful expansion of the exponential of the sum of two non-commuting matrices, one of which is diagonal

Christoph T. Koch, John Spence

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

The matrix exponential plays an important role in solving systems of linear differential equations. We will give a general expansion of the matrix exponential S = exp[λ(A + B)] as Sn,m = e λbndelta;n,m + ∑q=1 l1=0 N ⋯ ∑ lq-1=0 N alq-1,mCn,l1,...,lq-1,m (q) (B, λ) with Cn,l1,...,lq-1,m (q) (B, λ) being an analytical expression in bN, bl1, bl2, ... blq-1, bm, and the scalar coefficient λ A is a general N × N matrix with elements an,m and B a diagonal matrix with elements bn,m = bnδ n,m along its diagonal. The convergence of this expansion is shown to be superior to the Taylor expansion in terms of (λ[A + B]), especially if elements of B are larger than the elements of A. The convergence and possibility of solving the phase problem through multiple scattering is demonstrated by using this expansion for the computation of large-angle convergent beam electron diffraction pattern intensities.

Original languageEnglish (US)
Pages (from-to)803-816
Number of pages14
JournalJournal of Physics A: Mathematical and General
Volume36
Issue number3
DOIs
StatePublished - Jan 24 2003

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Matrix Exponential
expansion
matrices
Multiple Scattering
Taylor Expansion
Diagonal matrix
Electron Beam
Linear differential equation
Diffraction
Multiple scattering
Electron diffraction
Scalar
Diffraction patterns
Angle
Differential equations
differential equations
diffraction patterns
electron diffraction
Coefficient
scalars

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

A useful expansion of the exponential of the sum of two non-commuting matrices, one of which is diagonal. / Koch, Christoph T.; Spence, John.

In: Journal of Physics A: Mathematical and General, Vol. 36, No. 3, 24.01.2003, p. 803-816.

Research output: Contribution to journalArticle

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