# Change of coordinates resulting in scale

1. Nov 2, 2011

### miraiw

1. The problem statement, all variables and given/known data
Consider q: ℝ$^{2}$→ℝ
q(x,y) = $\left[x\ y\right] S \left[\stackrel{x}{y}\right]$

Show that with the change of coordinates
$\left[\stackrel{x}{y}\right] = Q\left[\stackrel{x'}{y'}\right]$
that q(x,y) =$\lambda_{1}x'^{2} + \lambda_{2}y'^{2}$

2. Relevant equations
$S = \left[\stackrel{2}{\sqrt{6}}\ \stackrel{\sqrt{6}}{-3}\right]$
$Q^{-1}SQ = \left[\stackrel{\lambda_{1}}{0}\ \stackrel{0}{\lambda_{2}}\right]$

3. The attempt at a solution
Found the eigen values of -4 and 3. Substituted Q<x', y'> for <x, y> everywhere in q(x,y) and got

q(x, y) = det(S) * ( λ1 x'2 + λ2 y'2)

Don't know where the scaling by det(S) comes in.

P.S.: Sorry, I have no idea how to format matrices with Latex.