- #1

iamalexalright

- 164

- 0

## Homework Statement

Let J be the nxn matrix all of whose entries are equal to 1. Find the minimal polynomial and characteristic polynomial of J and the eigenvalues.

Well, I figure the way I'm trying to do it is more involved then other methods but this is the easiest method for me to start with.

We can find the characteristic equation of J (and thus the eigenvalues) by computing:

[tex]det(J - \lambda I)[/tex]

We also know that the determinant of a nxn matrix can be found by computing:

[tex]det(A) = \sum_{\sigma \in S_{n}} sgn(\sigma) \prod_{i = 1}^{n}A_{i,\sigma_{i}}[/tex]

where S_n is the set of all permutations of (1,2,3,...n). sgn is the signature of the permutation.

Now we want to compute [tex]det(J - \lambda I)[/tex]

I know there will be n! terms after applying Leibniz's formula, half of which will have a positive signature and half that will have an odd signature.

Now, I don't really know how to prove this but with some investigation we can see that there will only be one term with degree n, ie:

[tex]det(J - \lambda I) = (1 - \lambda)^{n} + ...[/tex]

There shouldn't be any terms with degree n - 1...

I'm not sure where to go after this... I realize that this probably isn't the best method to go about this (it's late and I've been working on hw for awhile now...) so any other suggestions would be great (along with suggestions on how to continue with this path if possible).