To Find The Exponential Of a Matrix

Given a matrix $M=\begin{pmatrix} 1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1\end{pmatrix}$

My question is how can I find the exponential of M i.e. $e^M$?

I need the tricks to solve it. Is there any special method?

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We have $M^2=3M$, and by induction $M^n=3^{n-1}M$. This gives $$\exp M=\sum_{j=0}^{+\infty}\frac{M^j}{j!}=I+\sum_{j=1}^{+\infty}\frac{3^{j-1}}{j!}M=I+\frac{e^3-1}3M.$$

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For this matrix I think you can calculate $M^k$ explicitly. For example, $$M^2 = \left[ \begin{array}{ccc} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \\ \end{array} \right]$$ and $$M^3 = \left[ \begin{array}{ccc} 9 & 9 & 9 \\ 9 & 9 & 9 \\ 9 & 9 & 9 \\ \end{array} \right]$$ Generally, $M^k = 3^{k-1}M$. Plug that into the definition of the matrix exponential and you ought to be able to find a nice formula.

More generically, I like to calculate generalized e-vectors or use the Cayley Hamilton Theorem... there are many other methods.

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