Writing nonlinear ODE in matrix form I want to obtain the eigenvalues of the following nonlinear system
${\displaystyle {\dot {x}}_{1}(t)=x_{2}(t)}$ 
$  {\displaystyle{\dot {x}}_{2}(t)=-{\frac {g}{\ell }}\sin {x_{1}}(t)-{\frac {k}{m\ell }}{x_{2}}(t)}$
I have tried to convert to matrix form in order to find $A-I\lambda$
$A= [0, 1$
$-g/l\sin(x_1), -k/ml]$ 
But this doesn't seem right due to he sin term still being present, any advice?
 A: Normally what you do is to linearize the system about an equilibrium point, and then calculate eigenvalues and eigenvectors. Here, our vector field is $F: \Bbb{R}^2 \to \Bbb{R}^2$, defined by
\begin{align}
F(x_1,x_2) = \left(x_2, \,  -\frac{g}{l} \sin(x_1) - \frac{k}{ml}x_2  \right).
\end{align}
An equilibrium point is one where $F$ vanishes. It is easy to see that $F(0,0) = (0,0)$, so the origin is an equilibrium point. Now, let's compute the Jacobian matrix of $F$ at the origin:
\begin{align}
JF_{(0,0)} = 
\begin{pmatrix}
0 & 1 \\
-\frac{g}{l} & -\frac{k}{ml}
\end{pmatrix}
\end{align}
So, you can consider the linearised system
\begin{align}
x'(t) = JF_{(0,0)} \cdot x(t)
\end{align}
where for convenince I denote $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$. Now, you can definitely compute the eigenvalues and eigenvectors of $JF_{(0,0)}$.

You've been given a non-linear equation $x'(t) = F(x(t))$, so it doesn't make much sense to talk about eigenvalues of such a system. What you have done is to artificially (and incorrectly) "force" $F(x(t))$ into a product of a certain $2 \times 2$ matrix and the column vector $x(t)$, which conceptually isn't very meaningful. 
On the other hand, what I suggested is (usually) meaningful near the the equilibrium point, because you're approximating the non-linearity of $F$ by its derivative, which gives you a simple linear equation to analyze. And in general, knowing the behavior of the linear system tells you the behavior of non-linear system near the equilibrium point.
