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