- #1

war485

- 92

- 0

## 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 Q

^{T}AQ = 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 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?