Compute $e^M$ where $M=\begin{bmatrix}8 & -1\\4 & 4\end{bmatrix}$
Because M is not diagonalizable i try to use Jordan decomposition so i find the Jordan matrix to be $J=\begin{bmatrix}6 & 1\\0 & 6\end{bmatrix}$ but i cannot find the other matrix $S$. I found the eigenvector to be $V_1=\begin{bmatrix}1\\2 \end{bmatrix}$ then i try to find the generalized vectors for the eigenvalue $\lambda = 6$ but i'm doing something wrong on the way. Any help would be appreciated. Thanks
Best Answer
The generalized eigenvector you want is any solution to the equation $$ (M - 6I)x = v_1 $$ We can solve this equation by row-reduction: $$ \pmatrix{2&-1&1\\4&-2&2} \leadsto \pmatrix{2&-1&1\\0&0&0} $$ So, we can take for instance the generalized eigenvector $\pmatrix{0\\-1}$.
More generally, any vector of the form $t\pmatrix{0\\-1} + (1-t)\pmatrix{1/2 \\ 0}$ for $t \in \Bbb R$ will do.