There are some structures/forms that tend to surface while solving problems whose solutions are given in the form of eigen vectors and eigen values. Knowing some of those forms can help while trying to solve a larger problem because then you can pattern match the form to the solution. Here’s one pattern whose solution is given in the form of eigen vectors/values.

Maximizing a Quadratic Form

Let $ \um{M} $ be a symmetric matrix with real elements. The eigenvector of $ (M + M^\intercal)/2 $ that has the highest eigen value maximizes the following objective: