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Is it possible to define a Taylor expansion for matrices ? Can I use functional derivative ?

More precisely I have to calculate something like : $\ln(A+B)$ using a Taylor expansion, where $A$ and $B$ are hermitian matrices which depend also on $x\in \mathbb{R}^3$. My idea is to use functional derivative but I don't know if the result is correct!

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up vote 2 down vote accepted

In one particular case there is very good approach. Denote $C=A+B-I$, if $\operatorname{Sp}(C)\subset\mathbb{D}$ then we can use Tailor expansion for the $$ \ln(I+C)=\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n}C^n=\sum\limits_{n=1}^\infty\frac{(-1)^{n+1}}{n}(A+B-I)^n $$ If this conditions are not satisfied you should apply general theory - holomorphic functional calculus. In order to $\ln(A+B)$ make sense it is necessary that $\operatorname{Sp}(A+B)\subset\mathbb{C}^\times$, because $\ln\in\mathcal{O}(\mathbb{C}^\times)$. In this case we take arbitrary contour $\gamma$ containing $\operatorname{Sp}(A+B)$ and "simply" calculate $$ \ln(A+B)=\frac{1}{2i\pi}\int\limits_\gamma\ln(\lambda)(\lambda I-A-B)^{-1}d\lambda $$

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More explicitly: the series for the matrix logarithm is valid if and only if the eigenvalues of $C$ are within the disk of convergence of the usual scalar series for the logarithm... –  J. M. May 13 '12 at 8:53
    
As for the Cauchy-like approach indicated by the second formula in Norbert's answer, see this. –  J. M. May 13 '12 at 8:56
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