(adsbygoogle = window.adsbygoogle || []).push({}); 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 Q^{T}AQ = D

A =

[ 1 2 2 ]

[ 2 1 2 ]

[ 2 2 1 ]

2. Relevant equations

(A - [tex]\lambda[/tex]I ) = 0

3. 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 v_{1}= [ 1 1 1 ]

Then I'm almost done but I got stuck when trying to find v_{2}and v_{3}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 x_{1}, x_{2}and x_{3}are all free variables for v_{2}and v_{3}, but if that's the case, then how can I make v_{1}v_{2}v_{3}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?

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# Homework Help: Orthogonally diagonalizing the matrix

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