Here is a problem that you may enjoy thinking about. Given an matrix define a new matrix by the power series
This means, of course, the matrix whose entries are the limit of the corresponding entries of the sequence of matrices as
(This limit actually exists. Those of you who have seen Hilbert spaces should see a proof easily: Recall we defined the norm of as where in this supremum and denote the usual norm (of or respectively) in defined in terms of the usual inner product. One checks that a series of vectors converges (in any reasonable sense) in a Banach space if it converges absolutely, i.e., if converges. Since the series defining clearly converges absolutely.)
The matrix is actually a reasonable object to study. For example, the function is the unique solution to the differential equation Here, is a fixed vector.
Note that, for any the matrix is invertible, since as a direct computation verifies.
Anyway, the problem: Show that for any matrix we have Note this is not completely unreasonable to expect: A direct computation shows that if is an eigenvector of with eigenvalue then so the formula is true whenever is diagonalizable.