Question

In Exercises 57 through 61, consider a quadratic form q on ${\mathbf{R}}^{\mathbf{3}}$with symmetric matrix A, with the given properties. In each case, describe the level surface $\mathit{q}\left(\stackrel{\rightharpoonup }{x}\right)\mathbf{=}\mathbf{1}$geometrically.

60. qis indefinite and det A>0.

Short answer

The level curve represents two –sheeted hyperboloid.

Step-by-step solution

of 2: Given information

• Let us consider q(x,y,z) to be an indefinite quadratic form defined by a symmetric matrix B with det(B)>0.
• As per the theorem 8.2.2 , we know that the quadratic form q is diagonalizable with respect to the orthonormal eigenbasis of B
• Let us consider the diagonalized form of q with the diagonal matrix.

$\mathrm{A}=\left[\begin{array}{ccc}{\mathrm{\lambda }}_1& 0& 0\\ 0& {\mathrm{\lambda }}_2& 0\\ 0& 0& {\mathrm{\lambda }}_3\end{array}\right]$

• Here, A is the diagonal matrix, and hence the eigenvalues of A are the diagonal entries of $\mathrm{A},{\mathrm{\lambda }}_1,{\mathrm{\lambda }}_2,{\mathrm{\lambda }}_3$..

of 2: Application

• With respect to the theorem 8.2.4, we are aware that a matrix is a positive definite if and only if all its eigenvalues are positive.
• Similarly, a symmetric matric X is negative definite if and only if X is positive definite.
• Hence, we have all eigenvalues of a negative definite matrix as negative or non-positive.
• As A is indefinite, some eigenvalues must be positive and some must be negative too.
• As,$\det \mathrm{A}={\mathrm{\lambda }}_1.{\mathrm{\lambda }}_2.{\mathrm{\lambda }}_3>0$one of the eigenvalues must be positive, whereas all the other eigenvalues must be negative.
• Without the loss of generality, let${\mathrm{\lambda }}_1>0,{\mathrm{\lambda }}_2<0,{\mathrm{\lambda }}_3<0$, and so the quadratic form is.

$\mathrm{q}\left(\mathrm{x}\right)={\mathrm{x}}^{\mathrm{T}}\mathrm{Ax}=\left[\begin{array}{ccc}\mathrm{x}& \mathrm{y}& \mathrm{z}\end{array}\right]\left[\begin{array}{ccc}{\mathrm{\lambda }}_1& & \\ & {\mathrm{\lambda }}_2& \\ & & {\mathrm{\lambda }}_3\end{array}\right]\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]={\mathrm{\lambda }}_1{\mathrm{x}}^2+{\mathrm{\lambda }}_2{\mathrm{y}}^2+{\mathrm{\lambda }}_3{\mathrm{z}}^2$

• The quadratic form q(x)=1 is indefinite and level curve represents a two-sheeted hyperboloid.
• The quadratic form $\mathrm{q}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)={\mathrm{x}}^2-2{\mathrm{y}}^2-3{\mathrm{z}}^2$is indefinite with and the level curve .
• The level curve is sketched below:

Result:

The level curve represents two –sheeted hyperboloid.