Quadratic Surfaces

Curves in ${\mathbb{R}}^2$

Say we have a constant $C$ and $Q(x)=x^{T}Ax$ where $A\in {\mathbb{R}}^{2\times 2}$. Then, $C=x^{T}Ax$ Is a curve in ${\mathbb{R}}^2$. Say A=\b{2&1\\1&2}. Then $Q(x)=2x^2+2y^2+2xy=C$. If $C=1$ we can plot, Here is it plotted when we changed $C$ from 1 to 10.

Curves in ${\mathbb{R}}^3$

Say we have $z=Q(x)$ , z=x^2+y^2=x^T\b{1&0\\0&1}x We can graph this,

Definitions

Word	Definition	Eigenvalues
Positive Definite	If $Q>0$ for all $\vec{x}\ne 0$	all eigenvalues are positive
Negative Definite	If $Q<0$ for all $\vec{x}\ne 0$	all eigenvalues are negative
Positive Semidefinite	If $Q\ge 0$ for all $\vec{x}$
Negative Semidefinite	If $Q\le 0$ for all $\vec{x}$
Indefinite	If $Q$ takes on positive and negative values for $\vec{x}\ne 0$	at least one negative and one positive