Writing a system of conic equations

  • Thread starter Jhenrique
  • Start date
  • #1
685
4

Homework Statement


Given a system like: [tex]\\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[/tex] How write it in matrix form?

Homework Equations



The Attempt at a Solution


[tex]\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}[/tex]
I don't know how!
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,847
966
A single set of quadratic terms can be written
[tex]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}[/tex]

You should be able to tweak that to give two.
 
  • #3
D H
Staff Emeritus
Science Advisor
Insights Author
15,415
687
This works:

[tex]
\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}[/tex]

But why? What are you trying to do, Jhenrique?
 
  • #4
685
4
A single set of quadratic terms can be written
[tex]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}[/tex]

You should be able to tweak that to give two.

I can't see!

This works:

[tex]
\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}[/tex]

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.
 
  • #5
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
770
I think Halls means this:

A single set of quadratic terms can be written
[tex]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}[/tex]

You should be able to tweak that to give two.
 

Related Threads on Writing a system of conic equations

Replies
0
Views
1K
  • Last Post
Replies
8
Views
966
  • Last Post
Replies
1
Views
5K
  • Last Post
2
Replies
27
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
14
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
4
Views
2K
Top