# Matrix Mutiplication through Matrix Logarithms and Exponentials

Let $A,B$ be full rank $n \times n$ matrices. If $AB = BA$, then $\exp(\log(A)+\log(B))=AB$.

Supposing $A = USL$ and $B = VSL$ where $U,V,S,L$ are integer valued matrices, $det(L)=1$ and $U = LVL^{-1}$. If $AB = (USL)(VSL) = (L(VSL)L^{-1})(L^{-1}(USL)L) = (LBL^{-1})(L^{-1}AL)$, is there a possibility to use exponentials to calculate $AB$?

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Yes, the magic words are "Cambpell-Baker-Hausdorff", but why? –  Igor Rivin Aug 6 '12 at 17:34
Hi Igor: I thought the case here may be easier. I am trying to represent something that is commutative. But this is the best I am getting. –  J.A Aug 7 '12 at 3:52
Also would you happen to know the complexity of CBH. I have a long iterative chain of such matrices. –  J.A Aug 7 '12 at 3:56