[Tex/LaTex] aligning vector column

Question

I have the next latex code:

$$\begin{align*}\pi(X)f(x) = \frac{d}{dt}\Pi(e^{tX})\vert_{t=0} f(x) &= \frac{d}{dt}    
(\Pi(e^{tX})f(x))\vert_{t=0})= \\    
&=\frac{d}{dt}(f(e^{-tX}x))\vert_{t=0} & \\     
 &= \nabla_{\xi} f \l`enter code here`eft(-\[ \left( \begin{array}{ccc}    
X_{11} & \cdots & X_{1n} \\    
\vdots & \cdots & \vdots \\
X_{n1} & \cdots & X_{nn} \end{array} \right)\] \right) \[ \left( \begin{array}{ccc} \xi_1     
\\ \vdots \\ \xi_n \end{array} \right) \] &\\     
&= -\sum_{i,j}X_{ij}\xi_j \frac{\partial f}{\partial \xi_i} &\end{align*}$$

and the attached pic of what is shown:

My problems are:

1. there's a missing right bracket for the matrix with the minus sign symbol, how do I fix this?

2. the \xi's column vector is not placed at the right place it should be to right after the second right bracket which is missing from the output, and not placed above there, how to fix this?

3. the last equality should be aligned with the rest of the equalities i.e the equality should be placed beneath the rest of the equalities and not as it's displayed.

Any ideas as to how to fix this?

Thanks in advance, your help is much appreciated.

Answer 1

\begin{align*}
\pi(X)f(x) = \frac{d}{dt}\Pi(e^{tX})\vert_{t=0} f(x) &= \frac{d}{dt}(\Pi(e^{tX})f(x))\vert_{t=0})= \\
                                                     &=\frac{d}{dt}(f(e^{-tX}x))\vert_{t=0}\\ 
                                                     &= \nabla_{\xi} f \left(-
                                                            \left( \begin{array}{ccc}
                                                               X_{11} & \cdots & X_{1n} \\
                                                               \vdots & \cdots & \vdots \\
                                                               X_{n1} & \cdots & X_{nn}
                                                             \end{array} \right) 
                                                             \left( \begin{array}{c}  
                                                               \xi_1 \\ 
                                                               \vdots \\
                                                               \xi_n
                                                              \end{array} \right)
                                                      \right)\\ 
                                                      &= -\sum_{i,j}X_{ij}\xi_j \frac{\partial f}{\partial \xi_i}
\end{align*}