1. ## Matrix question

Hi. I need help with the final paer of this question. I am including my answers so far. Thanks in advance.

For the given martix A,
[-4 1 1]
[1 5 -1]
[0 1 -3]

a. Find the characteristic polynomail -60+x^3+2x^2-23x
b. Find the eigenvalues 5,-4,-3
c. Find the eigenvectors
[5] [1 1 10]
[-3] [8 0 -1]
[-4] [1 1 1]

The final part is where I need the help- or so I think!- If A is diagonalizable find a matrix P such that P^-1 AP is diagonal.
I just cannot figure it out.

2. You have done all the work. Simply make P the matrix with first column the eigenvector corresponding to the 1st eigenvalue, the second column the eigenvector corresponding to the second eigenvalue, ditto for the third.

This will give you the diagonal matrix with entries 1st eigenvalue, second eigenvalue, third eigenvalue when you do $P^{-1}AP$. This is in fact the Jordan Canonical Form of your matrix A, and it is unique up to permutation of the blocks (which in this case consist of just the eigenvalue).

So you can see in this way that you could put those columns in any order you want and you would just get a different diagonal matrix (ie just a permutation of the eigenvalues)