From: "Karl Forsberg" Subject: Re: MATRIX FUNCTIONS Date: Wed, 02 May 2001 17:34:27 GMT Newsgroups: sci.math.num-analysis Summary: [missing] "Michal Pitonak" skrev i meddelandet news:yvbbauhtlz2i@forum.mathforum.com... > Hi all! > I've got a serious problem with matrix exponential > function. > It's known, that if A,B don't commute then following > equality is false : exp(A).exp(B)=exp(A+B) > My question is : if A,B are antisymmetric matrixes, then > how can I find matrix C, such that: > exp(A).exp(B)=exp(A+B+C). > The result is known if [A,[A,B]] or [[A,B],B] = 0. > I would like to known the result for more general matrixes, i.e. > for any two antisymmetric matrixes > Thank's very much. What you want is essentially the Baker-Campbell-Hausdorff formula which gives C such that exp(A).exp(B)=exp(C) as an infinite series in terms of A, B and their commutators. It is guaranteed to always converge (and useful for practical purposes) only when A and B are in a nil-potent Lie-algebra such as in your example or eg for upper triangular matrixes. In this case the series is finite. For anti-symmetric 3x3 matrixes there is however an explicit finite expression: http://www.ii.uib.no/publikasjoner/texrap/abstract/2000-201.html Note that if your ultimate application is integration of Lie-group (rotation-matrix) valued ODEs there are specific and very efficient algorithms for this.