## Orthogonally diagonalizing the matrix

1. The problem statement, all variables and given/known data

This is for linear algebra/matrix:

Orthogonally diagonalize this matrix A by finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D

A =
[ 1 2 2 ]
[ 2 1 2 ]
[ 2 2 1 ]

2. Relevant equations

(A - $$\lambda$$I ) = 0

3. The attempt at a solution

D =
[5 0 0 ]
[0 -1 0 ]
[0 0 -1 ]

characteristic equation : -$$\lambda$$3 + $$\lambda$$2 + 9$$\lambda$$ + 5 = 0

$$\lambda$$ = 5, -1, -1 (I got these after factoring the characteristic equation)

when $$\lambda$$ = 5, I got v1 = [ 1 1 1 ]

Then I'm almost done but I got stuck when trying to find v2 and v3 when $$\lambda$$ = -1 because when I tried to do it, it turned out weird (it turned into a zero matrix!):
[ 0 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]

So I think it means that x1 , x2 and x3 are all free variables for v2 and v3 , but if that's the case, then how can I make v1 v2 v3 into an orthogonal matrix if they're not independent?!?! I almost got it but I've no idea what to do now! Does this mean that it is not possible to orthogonally diagonalize it?
 PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks
 Ok, I'm pretty sure I got it but I still have a problem.  sorry for making it complicated earlier. I'll dumb down my problem: I need help seeing that this matrix [ 1 1 1 ] [ 1 1 1 ] [ 1 1 1 ] have these two eigenvectors: [-1, 1, 0] [-1, 0, 1] how? I keep getting [ 0 -1 -1 ] and [ -1 0 -1 ]

Recognitions:
Gold Member
Staff Emeritus
 Quote by war485 have these two eigenvectors: [-1, 1, 0] [-1, 0, 1] how? I keep getting [ 0 -1 -1 ] and [ -1 0 -1 ]
It is impossible for a matrix to have exactly two eigenvectors. Instead, it might have a two-dimensional space of eigenvectors....

(incidentally, it's very easy to check if a given vector is an eigenvector...)

## Orthogonally diagonalizing the matrix

maybe I used the wrong terminology.
I think I meant that one of its eigenspace is the span of { [-1, 1, 0] , [-1, 0, 1] }
but I can't see how.

But I can see that its other eigenspace is [ 1 1 1 ]
 Recognitions: Gold Member Science Advisor Staff Emeritus First, I claim that it's very easy to show that that span is a subspace of the -1 eigenspace, just by direct verification. Secondly, I was trying to give you a hint by making you use more precise terminology. The problem is to find a particular vector space. Your answer key specified a basis for some vector space. Your work computed a basis for some vector space. You're focusing too much on the fact that your basis is different than the answer key's basis... but you haven't spent any effort checking whether or not the answer key's vector space is equal to or different from your vector space.... If you're given spanning sets for two vector spaces, how do you check if they're equal or not?
 Why orthogonally diagonalize a matrix?

Recognitions:
Gold Member