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 $\displaystyle A = UDU^{-1}$. 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 $\displaystyle A = UDU^{-1}$. 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?