Linear algebra is really all about linear transformations and the fundamental question is whether a matrix comes from some linear transformation with respect to some basis. In other words, are two matrices really from the same linear transformation? As proved above, this happens if and only if the two are similar. Canonical forms allow one to answer this question. There are two main kinds of canonical form, the Jordan canonical form for the case where the minimum polynomial splits and the rational canonical form in the other case. Of the two, the Jordan canonical form is the one which is used the most in applied math. However, the other one is also pretty interesting.