3.Inertia of Quadratic form

Definition3

The inertia(관성) of a symmetric matrix $A$ is a triple of integers denoted by  $\rm In(\it A)=(\it p, q, k)$ , where $p,q$ and  $k$ are the number of positive, negative and zero eigenvalues(고유값) of $A$, respectively.

$n=2)$

(n=2일때는 1. Conic section 의 interact와 Theorem1를 참고)

$n=3)$

Example) Detemine what is graph type of the given quadratic form.

1)  $q(x,y,z)=x^2+4xy-3y^2+z^2+5yz=6$

2)  $q(x,y,z)=2zy+zx=0$

Sol)

1) The matrix representaion of given quadratic form is $A=\begin{bmatrix} 1 & 2 &0 \\ 2 &-3 &\frac{5}{2} \\ 0 &\frac{5}{2} &1 \end{bmatrix}$.

The eigenvalues of $A$ are  $-\frac{1}{2}\sqrt{57}-1 <0, \frac{1}{2}\sqrt{57}-1 >0$ and $1>0$.

Thus, by above table the  $q(x,y,z)=x^2+4xy-3y^2+z^2+5yz=6$ is one sheeted hyperboloid.

2) The matrix representation of given quadratic form is $A=\begin{bmatrix} 0 &0 &\frac{1}{2} \\ 0 &0 &1 \\ \frac{1}{2} &1 & 0 \end{bmatrix}$ .

The eigenvalue of $A$ are $-\frac{1}{2}\sqrt{5} <0 ,\frac{1}{2}\sqrt{5}>0$ and $0$.

Thus, by above table the  $q(x,y,z)=2zy+zx=0$  is two palnes crossing in a line.

다음은 $n=3$ 이고 상수항( $c$ ) 이 0이 아닐때, quadratic form 의 형태를 표시해 주는 interact입니다.

다음은 $n=3$ 이고 상수항( $c$ ) 이 0일때, quadratic form의 형태를 표시해주는 interact입니다.

## Theorem3 (Sylvecter's Law of inertia for quadratic forms)

In the context of quadratic forms, a real quadratic form Q in n variables (or on an n-dimensional real vector space) can by a suitable change of basis (by non-singular linear transformation from x to y) be brought to the diagonal form

with each ai ∈ {0, 1, −1}. Sylvester's law of inertia states that the number of coefficients of a given sign is an invariant of Q, i.e. does not depend on a particular choice of diagonalizing basis( 기저 변환에 상관없이 양의 고유값, 음의 고유값 그리고 0인 고유값의 개수는 같다.). Expressed geometrically, the law of inertia says that all maximal subspaces on which the restriction of the quadratic form is positive definite (respectively, negative definite) have the same dimension. These dimensions are the positive and negative indices of inertia.

다음은 주어진 이차형식과 임의의 기저, $\alpha =\left \{ (1,0,1),(1,1,0),(0,0,1) \right \}$에 따른 변환된 행렬을 보여주고 그에 해당하는 고유값을 볼 수 있는 interact입니다.

(여기서 Sylvester's Law of inertia를 확인할 수 있습니다.)

Reference :

Definition3 & Examples : Linear Algebra, Kwark&Hong, Birkhauser, P282~p285

Theorem6 : sylvester'law of inertia, wikipidia, http://goo.gl/MBWBT9 , 2013,11,22

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