I Expanding a Potential Along the Principle Components

I have a potential given by ##V(x,y)## it is shaped like a squashed bowl and tilted, I'd like to expand the potential about the minima along the principal components, this should coincide with the smallest and largest confinement of the potential. To find the principal components I look at the curvature matrix, or the Hessian at the minimum

\begin{equation}
H=
\begin{bmatrix}
\partial_{x}^2V(x,y) & \partial_{xy}V(x,y)\\
\partial_{xy}V(x,y) & \partial_{y}^2V(x,y)
\end{bmatrix}
\end{equation}

I then want to find the eigenvectors of ##H## ##{v_1,v_2}##. The eigenvectors then correspond to the principal components which are directed along the axis of the smallest and largest confinement, these I would like to form a new basis from, with the unit vectors being ##{\hat{x'},\hat{y'}}##. The expression for the curvature matrix and eigenvectors is large so I haven't input them explicitly. What I want to do next is write my potential ##V(x,y)## as ##V(x',y')## and expand around either ##x',y'## to second order.

This is where I'm confusing myself, I'm simply struggling to understand how I write ##V(x,y)## as ##V(x',y')##? I think the transformation matrix should be
\begin{equation}
T=
\begin{bmatrix}
v_{1x}& v_{2x}\\
v_{1y} & v_{2y}
\end{bmatrix}
\end{equation}

But I'm not sure where to go from here?
 
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 

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