Portal de Periódicos da CAPES

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

2003; Institute of Physics; Volume: 36; Issue: 3 Linguagem: Inglês

10.1088/0305-4470/36/3/314

1361-6447

Christoph T. Koch, John C. H. Spence,

X-ray Spectroscopy and Fluorescence Analysis

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 = eλbnδn,m + ∑q = 1∞ ∑l1 = 0N ⋯ ∑lq−1 = 0Nan,l1 ⋯ alq−1,mC(q)n,l1,...,lq−1,m(B, λ) with C(q)n,l1,...,lq−1,m(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 = bnδn,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.