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Consider an indefinite quadratic form q on R3\mathbb{R}^{3} with symmetric matrix A. If det A < 0, describe the level surface q(x)=0.q(\vec{x})=0.


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Consider q(x,y,z)q(x,y,z) to be an indefinite quadratic form defined by a symmetric matrix BB with det(B)<0.(B)<0. By Theorem 8.2.2., we know that the quadratic form qq is diagonalizable with respect to the orthonormal eigenbasis of B.B.

Consider the diagonalized form of qq with the diagonal matrix

A=[λ1000λ2000λ3]A=\begin{bmatrix} \lambda_1 & 0 & 0\\0 & \lambda_2 & 0\\0 & 0 & \lambda_3 \end{bmatrix}

. Since AA is a diagonal matrix, thus the eigenvalues of AA are the diagonal entries of AA, viz., λ1\lambda_1, λ2\lambda_2 and λ3.\lambda_3.

From Theorem 8.2.4, we know that a matrix is positive definite (positive semi-definite) if and only if all its eigenvalues are positive (non-negative). Also, a symmetric matrix XX is negative definite (semi-definite) if and only if X-X is positive definite (positive semi-definite). Thus, we have all eigenvalues of a negative definite (semi-definite) matrix as negative (non-positive).

Recall that AA is indefinite so, by the previous discussion, some eigenvalues must be positive and some eigenvalues must be negative. Since detA=λ1λ2λ3<0A=\lambda_1 \cdot \lambda_2 \cdot \lambda_3<0, one of the eigenvalues must be negative, while the other eigenvalues must be positive. Without loss of generality let λ1>0\lambda_1 > 0, λ2>0\lambda_2 > 0, λ3<0\lambda_3 < 0.

Hence, the quadratic form

q(x)=xTAx=[xyz][λ1000λ2000λ3][xyz]=λ1x2+λ2y2+λ3z2q(\mathbf{x})=\mathbf{x}^TA\mathbf{x}=\begin{bmatrix} x & y & z \end{bmatrix} \begin{bmatrix} \lambda_1 & 0 & 0\\0 & \lambda_2 & 0\\0 & 0 & \lambda_3 \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix}=\lambda_1x^2+\lambda_2y^2+\lambda_3z^2

is indefinite and the level curve q(x)=0q(\mathbf{x})=0 represents an infinite elliptic cone.

The quadratic form q(x,y,z)=x2+2y2z2q(x,y,z)=x^2+2y^2-z^2 is indefinite with det(A)<0(A)<0 and the level curve q(x,y,z)=0q(x,y,z)=0 is an infinite elliptic cone, sketched below:

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