Linear Algebra, Quadratic Forms, Change of Variable (concept)

1. May 8, 2010

calvert11

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

Make a change of variable that transforms the quadratic form with no cross-product term:

9x1^2 - 8x1x2 = 3x2^2

2. Relevant equations

A = PDP^-1
Q = y^TDy

3. The attempt at a solution

I know the answer. This is a question regarding concept.

The eigenvalues for this problem are 1 and 11. The order in which I construct D affect the coefficients of the quadratic form following a change of variable.

Consider constructing D as either

1 0
0 11

or

11 0
0 1

The two resulting quadratic forms would have their coefficients switched.
Basically, I'm asking, is this ok?

Would both answers be acceptable?

2. May 8, 2010

lanedance

yes they are both ok, though the convention is usually to put the smaller eigenvalue first

moving the the diagonal coordinate system is change of basis to one composed of the eignevectors of the matrix A .

If you take the eigenvector corresponding to 1 as your first basis vector, and make it the first column of P then you will end up with the first form of D.

Alternately if you take the eigenvector corresponding to 11 as your first basis vector, and make it the first column of P then you will end up with the 2nd form of D.