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?
Oct 13th 2010, 06:23 PM
where is the ith eigenvalue and is the corresponding eigenvector. This is called the Jordan Decomposition of A.
Oct 13th 2010, 06:29 PM
Yes, but I don't understand which eigenvalue is lambda1, which is lambda2 etc. I'm sure I don't just pick randomly?
Oct 13th 2010, 06:31 PM
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.
Oct 13th 2010, 06:49 PM
So look at it this way
Which leads to .
So if you rearrange things you get the same result, but in a different order.
Oct 13th 2010, 06:58 PM
Ok, thanks. It's sounding like I did everything right then got cold feet at the end!