I have an equation: $VAV^T=B$
Where A and B are unknown 3x3 matrices. V is fully defined 3x3 matrix representing points in 3D:
$$\begin{matrix}
x1 & x2 & x3 \\
y1 & y2 & y3 \\
z1 & z2 & z3
\end{matrix}$$
To make it simpler, we can assume all z values equal 1.
How can I reframe this problem so that A and B can be represented?
A is a symmetric matrix of the form:
$$\begin{matrix}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c
\end{matrix}$$
B is also symmetric of the form:
$$\begin{matrix}
i^2 + x^2 & xy & x \\
xy & i^2 + y^2 & y \\
x & y & 1
\end{matrix}$$
Is there a way to solve for the 6 unknowns (a, b, c, i, x, y) - short of breaking this into a system of polynomial equations?