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)

 

[blue_raver22]

Thank you for your question. The problem with your method of finding the canonical form is that it may not always result in a unique solution. In other words, there may be multiple ways to manipulate the matrix to get a diagonal form, but only one of them will be the true canonical form. This is because the canonical form represents the unique characteristics of the quadratic form, and any other form may not accurately represent the same characteristics.

To ensure that you are finding the correct canonical form, you should pay attention to the following points:
1. Make sure you are using the correct operations for row/column manipulation. Using incorrect operations may lead to an incorrect solution.
2. Check if the resulting diagonal matrix has all 1's (or -1's for real canonical form) on the main diagonal. This is a necessary condition for the canonical form.
3. If there are multiple ways to manipulate the matrix to get a diagonal form, try to find a pattern or relationship between the different forms. This may help you identify the unique canonical form.
4. If you are still unsure about the uniqueness of the canonical form, you can use other methods such as completing the square or diagonalization to confirm your result.

In the example you provided, you are correct that the matrix can be manipulated to get different diagonal forms. However, the answer you provided as the canonical form is the only one that satisfies the necessary condition of having all 1's on the main diagonal. This indicates that it is the unique canonical form for this particular quadratic form.

I hope this explanation helps you better understand the concept of canonical form and how to find it accurately. If you have any further questions, please don't hesitate to ask. Keep up the good work in your studies!