Solve the exponential

Question

If $$ A = \begin{bmatrix} 3 & 4 \\ 4 & -3 \end{bmatrix}$$

Can someone find $\mathbf e^A$ ?

Edit 1:- I got this question in my fucntional analysis paper ! How is it even related to functional analysis ? Can somebody explain ?

Answer 1

$A$ is diagonalizable ! $A$ has the eigenvalues $ \pm 5$. Let $D:=\begin{bmatrix} 5 & 0 & \\ 0 & -5 & \end{bmatrix}$.

It is your turn to find an invertible matrix $P$ such that

$A=P^{-1}DP$.

Then $e^A=P^{-1}e^DP$.

$e^D$ is easy to compute.