The classical algorithm for solving is to write it as
which can have nonzero solutions **x** only if
. One shows inductively that
is a polynomial in , and thus one has `reduced' the eigenvalue problem to a polynomial rootfinding problem. This has theoretical advantages: one learns immediately that an **n** by **n** matrix produces a polynomial of degree **n** in and thus that every **n** by **n** matrix has **n** complex eigenvalues, counting multiplicity. We also learn that there will be no radical expression
for eigenvalues, in general, for . (There is, however, a
general expression for the **n=5** case in terms of elliptic functions;
this is widely regarded as not being useful---but maybe that will
be reconsidered).

But that's not the whole story, and indeed the case of multiple roots gets very complicated when we consider the eigenvectors **x**, and this
leads to the theory of the Jordan Canonical Form.

But in fact it turns out that neither reducing the eigenvalue problem to a polynomial problem nor computing the Jordan Canonical Form is a good idea if there are errors or uncertainties in your matrix, as we will see in the recap of numerical linear algebra. If we are working with a purely mathematical problem, though, and there are no uncertainties in the matrix whatever, then perhaps both these computations make sense.

Wed Jan 31 11:33:59 EST 1996