2
$\begingroup$

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?

$\endgroup$

2 Answers 2

7
$\begingroup$

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

$\endgroup$
3
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .