Finding the canonical form of a quadratic form.

[Omukara]

could someone please explain briefly what the problem is with my method of finding such canonical forms.

The method we've been taught is to find the canonical form by performing double row/column operations on the matrix representation of quadratic form until we get to a diagonal matrix, and manipulate the basis values by dividing to get so we get the desired form (i.e. in 1's (and -1's for real canonical form) however, my problem lies within understanding how this is unique?

Is there any other particulars aside from just doing operations on the matrix until I get a diagonal matrix I should pay attention to?

For instance the matrix;
0 0 1
0 1 0
1 0 0

could be manipulated to be the diagonal matrix;
1 0 0 0 0 0
0 1 0 0 -1 0
0 0 1 or 0 0 0, etc...

but the answer being;
1 0 0
0 1 0
0 0 -1

I can't comprehend why this is the unique canonical form. Any help would be much appreciated:)

 

[wisvuze]

The matrix is not unique; what *is* unique are the dimensions of the maximal size positive definite subspace, the radical, and the maximal size negative definite subspace. In other words, what is invariant are the number of 1's, -1's and 0's appearing on the diagonal of your matrix -- not that the matrix will always be the same.
The unique number of 1's, -1's is called the signature of the form

This result follows from http://en.wikipedia.org/wiki/Sylvester's_law_of_inertia

 

[Bacle]

But this transformation --and being able to diagonalize, assumes that the matrix rep.
of the form is non-singular. What guarantees this?

 

[Omukara]

Wisvuze, thanks!

After reading your comment I attempted the question again and saw my mistakes immediately. I can only blame it on the fact that it was past 1 in the morning:) I'm aware of Sylvester's Law of Inertia - however, I couldn't get the fact it had a unique rank/signature since I kept getting incorrect numbers - but of course it was due to my use of miracle row/column operations:PBacle, if the matrix representation of the form wasn't non-singular, it would't be diagonalisable - so we wouldn't be able to reduce it to canonical form (I believe:P)