Math Help - quadratic equations in matrix form.

1. quadratic equations in matrix form.

Hi

I'm sure this question will seem seem very trivial, but I was wondering if you could help me out.

Although I can see that quadratic equations such as

5x^2 -6xy+5y^2=8 can be shown as the matrix (xy)(5 -3/-3 5)=8 (Using the divide symbol to indicate row down.)

Similarly

7x^2-12xy-2y^2=10 would be (x y)(7 -6 /-6 -2)

but I was stuck when trying to convert the quadratic equation x^2 + x-8 +5xy -6y +2y^2=0 into matrix form.

If someone could shed some knowledge on the subject I would be deeply appreciative.

Thanks

Matt

2. Hello,

That's something about quadratic forms. If I'm not mistaking, the third one isn't one, so you can't find the matrix form of this equation...

Also, don't forget $\begin{pmatrix} x\\ y\end{pmatrix}$ on the right side of the 2x2 matrix...

3. Hi there, thanks for the response.

Then reason I asked was because I am trying to transform the last equation so that it can be expressed in the form
AX^2 + BY^2=1 and was trying to do this using eigen analysis, and I'm not sure if this is possible without matrices?
Do you know of any other ways I could set about doing this?

Thanks
Matt

4. Actually, you can do it, but it's a bit tricky, and it has to be with shifted variables. You'd like to write your equation this way:

$(x+a)^{2}+5(x+a)(y+b)+2(y+b)^{2}=C.$

Multiplying this out and comparing coefficients with your original equation lead to the following three equations:

$2a+5b=1$

$5a+4b=-6$

$C=8+a^{2}+5ab+2b^{2}.$

You can solve this system rather straight-forwardly. Then you can write your quadratic form as the following:

$\begin{bmatrix}x+a&y+b\end{bmatrix}\begin{bmatrix} 1 &5/2\\
5/2 &\sqrt{2}\end{bmatrix}\begin{bmatrix}x+a\\ y+b\end{bmatrix}=C.$

This may or may not be allowed, depending on your definitions, as Moo pointed out. But this computation can still be done.