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\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?