
diagonalized matrices
Hi guys, I'm hoping you can help me. I'm sure I've done something wrong, though I can't figure out what. The question I have to answer is this:
Find invertible U and diagonal D such that . A is the matrix:
2, 4, 2
4, 2, 2
2, 2, 1
I have found the three eigenvalues 2, 2, 7. I understand that these values, along the diagonal of a matrix otherwise filled with zeros, make up the D matrix.
I have also found the eigenvectors that go with the eigenvalues. Respectively they are [0,1,2], [1,0,2], [1,1,1/2]. I understand that these three vectors together make up U.
My problem is how to fit it all together. I don't understand what order I have to put the vectors in within U, or the values in within D, to prove that . I've tried to brute force it but I'm not getting anywhere and it's making me think I've done everything wrong.
Can anyone explain my mistake, please?

Define
and then
where is the ith eigenvalue and is the corresponding eigenvector. This is called the Jordan Decomposition of A.

Yes, but I don't understand which eigenvalue is lambda1, which is lambda2 etc. I'm sure I don't just pick randomly?

Well as long as you are consistent it it doesn't matter (number of eigenvalues is meaningless anyway). You just have to make sure that column 1 of matrix U corresponds to the eigenvector for the first eigenvalue in D, etc.

So look at it this way
Which leads to
.
So if you rearrange things you get the same result, but in a different order.

Ok, thanks. It's sounding like I did everything right then got cold feet at the end!