Canonical Form for quadratic equations *with* linear terms

In summary, the canonical form for a quadratic equation in two (or more) variables is either the first form (using the orthogonal matrix Q), or a form with translated coordinates.
  • #1
arestes
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?
 
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  • #2
That's a neat condition (I haven't checked your work). ##c^2-4ab=0## means that ##D## is going to be singular, which people often want to exclude when talking about quadratic forms anyway.

In particular, suppose your quadratic form was ##x^2+y=0##. It's pretty clear you're not going to change coordinates to make this look nice, since it's just not quadratic in one direction.
 
  • #3
My personal preference is the first one (simple scalar).
 
  • #5
WWGD said:
I think you're referring to Quadratic forms:
https://en.wikipedia.org/wiki/Quadratic_form
Yeah but quadratic forms don't have linear terms.

Thanks for reminding me of wikipedia. I did find the info I needed (almost) here:https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections

I found that there is a general form to write the equation of a conic (valid for central conics, which excludes parabolas because the K mentioned below doesn't exist):
1623477016440.png


Here they call this equation "standard canonical form" but I'm not sure if it's THE canonical form.
My question still stands regarding the naming convention but I just realized that it's just that: semantics.

So, in the end (after learning stuff about projective geometry and homogeneous coordinates) is the name "canonical form" used for the "standard canonical" form above (although it doesn't work for parabolas for which, I guess it'll be just y^2=ax or x^2=ay)?
 

Related to Canonical Form for quadratic equations *with* linear terms

1. What is the purpose of using canonical form for quadratic equations with linear terms?

The purpose of using canonical form for quadratic equations with linear terms is to simplify and standardize the equation, making it easier to solve and analyze. It also helps in identifying the key characteristics of the equation, such as the vertex, axis of symmetry, and roots.

2. How do you convert a quadratic equation with linear terms into canonical form?

To convert a quadratic equation with linear terms into canonical form, you need to complete the square by adding or subtracting a constant term to both sides of the equation. This constant term is calculated by taking half of the coefficient of the linear term, squaring it, and then adding or subtracting it to the constant term of the original equation.

3. Can all quadratic equations with linear terms be converted into canonical form?

Yes, all quadratic equations with linear terms can be converted into canonical form by completing the square. However, the process may be more complex for equations with higher degree terms or non-standard coefficients.

4. How does canonical form help in solving quadratic equations with linear terms?

Canonical form simplifies the equation, making it easier to solve by hand or with a calculator. It also helps in identifying the key characteristics of the equation, such as the vertex and roots, which are important in solving and graphing the equation.

5. Are there any disadvantages to using canonical form for quadratic equations with linear terms?

One potential disadvantage of using canonical form is that it may not be as intuitive to understand as the standard form of a quadratic equation. It also may not be as useful in certain applications, such as when dealing with complex numbers or when using the quadratic formula.

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