I've been at this for a while and I'm now stumped.

The matrix, A is

5 -4 0

-4 3 -4

0 - 4 1

Here's what I've got so far.

eigenvalues are λ = 9, λ = -3, λ = 3

corresponding eigen vectors are (2k, -2k, k), (k/2, k, k), (-k, k/2, k)

{(2, -2, 1)} is a basis for s(9)

{(1,2,2)} is a basis for s(-3)

{(-2,-1,2)} is a basis for s(3)

So that gives the set

E = {(2,-2,1),(1,2,2),(-2,-1,2)} which is an eigenvector basis of A (not sure if this is even needed as I think I need an orthonormal basis)

I think I need to find an orthonormal eigenvector basis next, but I can't understand how to do that (if that is actually next). As that should give me the transition matrix.

So looking for someone to hold my hand to the end of this process please