1.Conic Section

Definition1

1) An equation of the form

$f(\rm x)=\it \sum _{i=1}^{n} a_{ij}x_{i}x_{j}+\sum _{i=1}^{n}b_{i}x_{i}+c=\rm 0$

where $a_{ij}$ , $b_{i}$ and  $c$ are real constants, is calles a quadratic equation(이차방정식) in $n$ variable $x_{1} , x_{2}, \cdots ,x_{n}$.

In matrix form, it can be written as

$f(\rm x)=x^{\it T}\it A\rm x +b^{\it T}\rm x +\it c=\rm 0$,

where  $A=[a_{ij}]$, $\rm x =[\it x_{1}\cdots x_{n}]^{T}$ and  $\rm b =[\it b_{1}\cdots b_{n}]^{T}$ in $\mathbb{R}^n$

2)  A linear form(일차형식) is polynimial of degree 1 in $n$ variables $x_{1}, x_{2}, \cdots ,x_{n}$ of form

$\rm b^{\it T}\rm x = \it \sum _{i=1}^{n} b_{i}x_{i}$

3) A quadratic form(이차형식) is a (homogeneous) polynimial of degree 2 in $n$ variables $x_{1}, x_{2}, \cdots ,x_{n}$ of form

$q(\rm x)=\rm x^{\it T}\it A\rm x=\it [x_{1} \cdots x_{n}][a_{ij}]\begin{bmatrix}x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix}=\sum_{i=1}^{n} \sum_{j=1}^{n}a_{ij}x_{i}x_{j}$

4)  From the definition of a quadratic form, one can see that, fixing a basis like the standard basis for $\mathbb{R}^{n}$, a quadratic form is associated with a unique symmetric matrix, which is called the matrix representation(행렬 표현) of the quadratic form $q$ with respect to the basis chosen.  On the other hand any (real) symmetric marix $A$ gives rise to a quadratic form $\rm x^{\it T}\it A\rm x$.

$*$ Remark

Any square matrix $A$ is sum of a symmetric matrix  part $B$ and skew-symmetric matrix part $C$,

$A=B+C$   ,    where   $\small B=\frac{1}{2}(A+A^{T})$  ,  $\small C=\frac{1}{2}(A-A^{T})$

For the skew-symmetric matrix $\small C$, we have

$\small \rm x^{\it T}\it C\rm x=(\rm x^{\it T}\it C\rm x)^{\it T}=\rm x^{\it T}\it C^{T}\rm x=-\rm x^{\it T}\it C\rm x$.

Hence, as a real number, $\small \rm x^{\it T}\it C\rm x=0$. Therefore,

$\small q(\rm x)=\rm x^{\it T}\it A\rm x=\rm x^{\it T}\it (B+C)\rm x=\rm x^{\it T}\it B\rm x$.

This means that, whitout loss of generality, one may assume that the matrix $\small A$ in definition of a quadratic form is a symmetric matrix(대칭행렬).

Definition2

Conic sections have the form of a second-degree polymomial:

That can be written as:

where  is the homogenous coordinate vector:

And  a matrix representation of $\small Q(\rm x)$:

Theorem1

If , the conic is degenerate.

In mathematics, a degenerate conic is a conic that fails to be an irreducible curve.

In the real plane, a degenerate conic can be two lines that

1. may or may not be parallel, 2. a single line (actually two coinciding lines), 3. a single point, 4.or the null set (no points).

If Q is not degenerate, we can see what type of conic section it is by computing the minor  (that is, the determinant of the submatrix resulting from removing the last row and the last column of AQ):

• If and only if , it is a hyperbola(쌍곡선).
• If and only if , it is a parabola(포물선).
• If and only if , it is an ellipse(타원).

Example) Sketch the graph of  $f(x,y)=9x^2+4y^2=144$ and determine what it is type.

(참고 :http://matrix.skku.ac.kr/2013-Album/SKKU-LA-SolutionsSPF/Image545.jpg.2013.12.02)

Sol)

다음은 2개의 변수에 대한 이차형식의 형태가 어떤 모양인지를 보여주는 interact입니다.

(3. inertia of quadratic form에서도 활용됩니다.)

Reference :

Definition1- Linear Algebra, Kwak & Hong, Birkhauser, p280~p281.

Theorem2- Linear Algebra, Kwak& Hong, Birkhauser, p294.

Definition2, Theorem1- Matrix representation of Conic section, wikipedia, http://goo.gl/NMNZfV ,2013.11.25

예제와 그림파일 :이상구, http://matrix.skku.ac.kr/2013-Album/SKKU-LA-Solutions.html, 2013. 11. 26

Back TO the HOME