Writing a system of conic equations

[Jhenrique]

Homework Statement

Given a system like: $$\\a x^2 + b xy + c y^2 + d x + e y + f = 0 \\g x^2 + h xy + i y^2 + j x + k y + l = 0$$ How write it in matrix form?

Homework Equations

The Attempt at a Solution

$$\begin{matrix} a x^2 + b xy + c y^2\\ g x^2 + h xy + i y^2\\ \end{matrix} + \begin{bmatrix} d & e\\ j & k\\ \end{bmatrix}\begin{bmatrix} x\\ y\\ \end{bmatrix} + \begin{bmatrix} f\\ l\\ \end{bmatrix} = \begin{bmatrix} 0\\ 0\\ \end{bmatrix}$$
I don't know how!

 

[HallsofIvy]

A single set of quadratic terms can be written
$$ax^2+ bxy+ cy^2= \begin{bmatrix}x & y \end{bmatrix}a & b/2 \\ b/2 & c \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$$

You should be able to tweak that to give two.

 

[D H]

This works:

$$\begin{bmatrix} x & y & 0 & 0 \\ 0 & 0 & x & y \end{bmatrix} \begin{bmatrix} a & b/2 \\ b/2 & c \\ g & h/2 \\ h/2 & i \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} d & e \\ j & k \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}+ \begin{bmatrix} f\\ l \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix}$$

But why? What are you trying to do, Jhenrique?

 

[Jhenrique]

A single set of quadratic terms can be written
$$ax^2+ bxy+ cy^2= \begin{bmatrix}x & y \end{bmatrix}a & b/2 \\ b/2 & c \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$$

You should be able to tweak that to give two.

I can't see!

This works:

$$\begin{bmatrix} x & y & 0 & 0 \\ 0 & 0 & x & y \end{bmatrix} \begin{bmatrix} a & b/2 \\ b/2 & c \\ g & h/2 \\ h/2 & i \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} d & e \\ j & k \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}+ \begin{bmatrix} f\\ l \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix}$$

But why? What are you trying to do, Jhenrique?

I'm trying to write a vectorial equation that represents a system not of linear equations but yes of quadratic equations!

If a linear equation (or a affim equation) can be represent a linear system, so, a quadratic equation can represent a quadratic system.

 

[LCKurtz]

I think Halls means this:

A single set of quadratic terms can be written
$$ax^2+ bxy+ cy^2= \begin{bmatrix}x & y \end{bmatrix}\begin{bmatrix}a & b/2\\b/2 & c \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}$$

You should be able to tweak that to give two.