Expert: Paul Klarreich - 4/3/2006

Question
Find principal axes of the quadratic form
q(x) = 2(x1)^2 + 3(x2)^2 + 2(x3)^2 - 2x1x3

Thank You
Nadia  

Answer
Hi, Nadia,
I am afraid it has been a long time since I did any linear algebra and it may take me some time to work this out.

However, I think you will start by finding eigenvalues, something like this, perhaps:

For your form:
q(x) = 2(x1)^2 + 3(x2)^2 + 2(x3)^2 - 2x1x3

The symmetric matrix is:

[  2   0   -1 ]
[  0   3    0 ]
[ -1   0    2 ]


The determinant is

2(6) - 0(..) - 1(3) = 9


Finding the eigenvalues:

[ L-2    0    -1 ]
[    0  L-3    0 ]
[   -1   0   L-2 ]

(L-2)^2(L-3) - (L-3) = 0
(L-3)[(L-2)^2 - 1] = 0

L = 3

L-2 = +-1
L = 2+-1
L = 1
L = 3

So the eigenvalues are  1,2,3
If I can go back and recall more, I'll follow up.