I've been trying to figure out how to find the density contour for a multivariate normal density function with an arbitrary number of dimensions. I've found a lot of examples for 3Dimensions and for 2Dimensions and several others (I can't put more links here:( ) and they always associate the biggest eigen vector and value to the biggest axis in the ellipse which makes sense, but they also associate them in the same order to the ellipse equation, i.e. the biggest length is associated to $a$ and the smaller to $b$
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
Why is this? And does this happen the same way with more dimensions? Does it require an order from the biggest to the smaller? That is associated consecutively? a -> Biggest b -> 2nd Biggest c -> 3rd Biggest ....
I've also seen two ways of finding the axis length to plug into a,b,c...Actually they are the same with a small difference. The $\chi _{2}^{2}$ is the chi-square distribution, used to specify the % of area/surface that the ellipse will encompass
$$
a = \sqrt{\chi _{2}^{2}} \sqrt{\lambda _{i}}
$$ or $$
a = 2\sqrt{\chi _{2}^{2}} \sqrt{\lambda _{i}}
$$
Which is the correct?
I'm intersecting the ellipsoid eq. with a line equation, and it is really important that the intersection occurs in the correct place :/
Thanks!