- #1

- 80

- 3

- TL;DR Summary
- Not sure about the accepted "canonical form" for a quadratic equation WITH linear term

Hello:

I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables:

$$ax^2+by^2+cxy+dx+ey+f=0$$

Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part):

$$ w^TDw+[d \ \ e]w+f=0$$

$$w^TDw+Lw+f=0$$

where

$$ w=\begin{pmatrix}

x' \\

y'

\end{pmatrix} = Q^T

\begin{pmatrix}

x \\

y

\end{pmatrix}

$$

and

$$ L=[d \ \ e] $$

Or is it a form with translated coordinates:

$$a(x'-m)^2+b(y'-n)^2+c(x'-m)(y'-n)+d(x'-m)+e(y'-n)+f'=0$$

with some to-be-determined constants m and n such that the linear terms vanish, which can be then used to change variables x'=x+m and y'=y+n.

I tried to find these m and n (expanding the binomials) but the simultaneous equations to satisfy in order to remove the linear terms are restricting and it seems to be impossible when $$c^2-4ab=0$$

Is it enough to leave the linear terms and call it "canonical form" just by diagonalizing the quadratic terms?

I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables:

$$ax^2+by^2+cxy+dx+ey+f=0$$

Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part):

$$ w^TDw+[d \ \ e]w+f=0$$

$$w^TDw+Lw+f=0$$

where

$$ w=\begin{pmatrix}

x' \\

y'

\end{pmatrix} = Q^T

\begin{pmatrix}

x \\

y

\end{pmatrix}

$$

and

$$ L=[d \ \ e] $$

Or is it a form with translated coordinates:

$$a(x'-m)^2+b(y'-n)^2+c(x'-m)(y'-n)+d(x'-m)+e(y'-n)+f'=0$$

with some to-be-determined constants m and n such that the linear terms vanish, which can be then used to change variables x'=x+m and y'=y+n.

I tried to find these m and n (expanding the binomials) but the simultaneous equations to satisfy in order to remove the linear terms are restricting and it seems to be impossible when $$c^2-4ab=0$$

Is it enough to leave the linear terms and call it "canonical form" just by diagonalizing the quadratic terms?