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

Christoph T. Koch, John Spence

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14 Scopus citations


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=0N ⋯ ∑ lq-1=0N 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
Issue number3
StatePublished - Jan 24 2003


ASJC Scopus subject areas

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

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