- #1

Apothem

- 39

- 0

Moved from technical forums, so no template

Problem:

Express the quadratic form:

q=x

in canonical form using Lagrange's Method/Algorithm

Attempt:

Not really applicable in this case due to the nature of my question

The answer is as follows:

Using the change of variables:

x

x

x

Indeed you get q'=(y

I can see that when you substitute it in it does indeed reduce it to canonical form however my question is as follows:

How am I meant to arrive at the above change of variables? I have seen a few cases of completing the square, but to my knowledge we can't use that here. EDIT: We can

Any help is greatly appreciated!

EDIT:

I think I've been able to deduce it...

x

Express the quadratic form:

q=x

_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}in canonical form using Lagrange's Method/Algorithm

Attempt:

Not really applicable in this case due to the nature of my question

The answer is as follows:

Using the change of variables:

x

_{1}=y_{1}+y_{2}x

_{2}=y_{1}-y_{2}x

_{3}=y_{3}Indeed you get q'=(y

_{1}+y_{3})^{2}-(y_{2})^{2}-(y_{3})^{2}I can see that when you substitute it in it does indeed reduce it to canonical form however my question is as follows:

How am I meant to arrive at the above change of variables? I have seen a few cases of completing the square, but to my knowledge we can't use that here. EDIT: We can

Any help is greatly appreciated!

EDIT:

I think I've been able to deduce it...

x

_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}=(x_{3})^{2}+x_{3}(x_{2}+x_{1})-(x_{3})^{2}and from there I completed the square, the only thing I'm still a bit uncertain on is that I worked this out the other way i.e. I worked out y in terms of x and not x in terms of y...
Last edited: