2.Eliminate Cross-product term

A quadratic form may be rewritten as the sum of two parts :

in which the frist part    is called the (perfect) square terms and the second part   is called the cross-product terms(교차곱). Actually, what makes it hard to sketch the quadratic surface is the cross product terms. However, the symmetric matrix  can be orthogonally diagonalized(직교 대각화), i,e., there exists an orthogonal matrix   such that

.

Here, the diagonal entries  's are the eigenvalues of  and the column vectors of  are their associated eigenvectors of . Then we get, by setting  ,

,

which is a quadratic form without the cross-product terms. Consequently, we have proven the following theorem.

Theorem 2 (Principal axis Theorem: 주축정리)

Let   be a quadratic form in    for a symmetric matrix . then there is a change of coordinate of  into   such that  .

,

where  is an orthogonal matrix(직교행렬) and

Example)  Eliminate the cross-product terms in the conic section  . This equation can be written in the form

The matrix    has eigenvalues  with associated eigenvectors    and

the unit vectors    and .

Then change coordinates,    for    . i,e,

다음은 이변수 이차형식에 대해 주축정리에 의해 교차곱을 없앤 이차형식의 형태와 그 행렬을 나타내주는 interact입니다.

Example)  Eliminate the cross-product terms in the conic section  . The matrix for the given quadratic form is

The eigenvalues  of  are   and associated eigenvectors are   .

Hence, orthogonal matrix  that diagonalizes  is

Thus, the equation is transformed to , which is a hyperbolic cylinder.

다음은 3변수에 이차형식에 대해 주축정리에 의해 교차곱이 없어진 형태와 그 행렬을 나타내주는 interact입니다.

좀더 자세한 내용 (참조: http://math1.skku.ac.kr/home/pub/751/ 2013.11.27)

http://matrix.skku.ac.kr/2013-Album/SKKU-LA-SolutionsSPF/Image546.jpg2013.11.27 http://matrix.skku.ac.kr/2013-Album/SKKU-LA-SolutionsSPF/Image547.jpg2013.11.27 http://matrix.skku.ac.kr/CLAMC/chap8/Page41.htm 2013.11.27

3. 선형대수학 Section 8-4, Quadratic Ft

http://youtu.be/lznsULrqJ_0 2013.11.27

Reference:

Diagonalization of quadratic form, Theorem5 & Examples : Linear Algebra, Kwark&Hong, Birkhauser, P282~p285

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