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

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[?(A + B)] as Sn,m = e?bn?n,m + ?q = 1? ?l1 = 0N ? ?lq?1 = 0Nan,l1 ? alq?1,mC(q)n,l1,...,lq?1,m(B, ?) with C(q)n,l1,...,lq?1,m(B, ?) being an analytical expression in bn, bl1, bl2, ... blq?1, bm, and the scalar coefficient ?. A is a general N ? N matrix with elements an,m and B a diagonal matrix with elements bn,m = bn?n,m along its diagonal. The convergence of this expansion is shown to be superior to the Taylor expansion in terms of (?[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.