- #1
war485
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Homework Statement
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 ]
Homework Equations
(A - [tex]\lambda[/tex]I ) = 0
The Attempt at a Solution
D =
[5 0 0 ]
[0 -1 0 ]
[0 0 -1 ]
characteristic equation : -[tex]\lambda[/tex]3 + [tex]\lambda[/tex]2 + 9[tex]\lambda[/tex] + 5 = 0
[tex]\lambda[/tex] = 5, -1, -1 (I got these after factoring the characteristic equation)
when [tex]\lambda[/tex] = 5, I got v1 = [ 1 1 1 ]
Then I'm almost done but I got stuck when trying to find v2 and v3 when [tex]\lambda[/tex] = -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?