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The problem of finding and classifying all relative equilibrium configurations of $N$-point vortices in the plane is first described as
a classical variational principle and then formulated as a
problem in
linear algebra. Given a configuration of $N$ points in the plane, one must
understand the structure of
the
$N(N-1)/2 \times N$ configuration matrix $A$ obtained by requiring that all interparticle distances
remain fixed in time.
If the determinant of the square, symmetric $N \times N$ covariance matrix
$A^T A$ is zero, there is a non-trivial nullspace of $A$ and
a basis set for this nullspace
can be used to determine all vortex strengths $\vec{\Gamma} \in R^N$
for which the configuration remains rigid.
Optimal basis sets are obtained
by using the singular value decomposition of $A$ which allows one
to categorize exact equilibria, approximate equilibria, and the
distance between different equilibria in the appropriate vector
space, as characterized by the Frobenius norm.