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I have a $k$-dimensional ellipsoid defined as

$\left(x-\mu\right) C \left(x-\mu\right)'<1$.

As an example assume $k=3$ (in practice my $k>100$) and

$ C = \begin{pmatrix} c_{1,1} & c_{1,2} & c_{1,3} \\ c_{2,1} & c_{2,2} & c_{2,3} \\ c_{3,1} & c_{3,2} & c_{3,3} \\ \end{pmatrix}$

I want to plot a two-dimensional view of it, say the first two of the $k$ dimensions. How does the $C'$ of the reduced ellipsoid look that I need to plot? I guess its not just

$ C' = \begin{pmatrix} c_{1,1} & c_{1,2} \\ c_{2,1} & c_{2,2} \\ \end{pmatrix}$

at least the resulting ellipse looks very big compared to points I draw uniformly from within the high-dimensionsinal ellipsoid (and then look at the plot of the points' first two dimensions).

Thanks for any hint!

Arctic Char
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Joe1979
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  • $C'$ would be a slice of $C$. In infer from your claims that you want to draw a projection of $C$. – Arnaud Oct 07 '20 at 12:23
  • See my answer to https://math.stackexchange.com/questions/3853351/do-ellipsoids-cast-ellipsoidal-shadows – Arnaud Oct 07 '20 at 12:27
  • Thanks for the hint I do look for a projection @Arnaud, I did not realise that at first. I tried to follow the solution (answer) here: link. However, plotting $\mathbf{(x_{1-2}-\mu_{1-2})} \left( J - L'K^{-1}L\right)\mathbf{(x_{1-2}-\mu_{1-2})}= 1$ gave me a hugh ellipse, but maybe I missundertand something and the formula does not give the desired projection. – Joe1979 Oct 07 '20 at 14:23
  • Dunno, but the projections of a 100 dimensional random ellipsoid are much bigger than their sections, so it seems ok. – Arnaud Oct 07 '20 at 14:31

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