7
$\begingroup$

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup.

I went through a theorem in the book stating the relationship between ellipsoid's radii and the eigenvalues.

So the ellipsoid is defined as following:

$\sum_i \sum_j (x_i - u_i)(x_j-u_j)c_{ij} \leq k$

where $C=[c_ij]$ is an symmetric positive definite matrix, $x, u$ are n-dimensional vectors. The book says that the radii of this ellipsoid is $r_i = \frac{k}{\sqrt{\lambda_i}}$ where $\lambda_i$ are the eigenvalues of matrix C. However it does not give any proof for that. I've tried to google for the proof but didn't get anything useful. Could someone please help me understand it?

Thanks,

  • 4
    You know that a symmetric positive definite matrix $\mathbf A$ can be decomposed as $\mathbf A=\mathbf V\mathbf \Lambda\mathbf V^\top$, $\mathbf V$ being an orthogonal matrix and $\mathbf \Lambda$ being the diagonal matrix of eigenvalues?2011-11-08
  • 0
    I think $A=V\Lambda V^{-1}$ instead of $V^T$, right? But how do you use it to prove?2011-11-08
  • 0
    I mentioned $\mathbf V$ is orthogonal, no?2011-11-08
  • 0
    Oh, sorry. I think I got it. So, $x^TAx=x^TV\Lambda V^Tx=y^T\Lambda y=\sum_i \lambda_i y_i^2$, right? Thanks.2011-11-08
  • 0
    I wonder why @Heike deleted their answer. I would have upvoted it.2013-01-15

2 Answers 2