A $k$-dimensional ellipse, surface and interior, with axes along the standard coordinates is algebraically defined as the set of points $z = (z_1, \ldots , z_k)^T$ satisfying $(z_1/\alpha_1)^2 + \cdots + (z_k/\alpha_k)^2 \leq 1$.
We can have a $k$-dimensional ellipse embedded inside $\mathbb{R}^n$ even in the case $n > k$ by allowing some of the $z_j$ to be identically zero. Using these definitions, show that the matrix $\Sigma = \text{diag}(\sigma_1,...,\sigma_{\min(m,n)}) \in \mathbb{R}^{m \times n}$, where $\sigma_1 \geq \ldots \geq \sigma_{\min (m,n)} \geq 0$, maps the unit sphere $\{x \in \mathbb{R}^n : ||x||_2 \leq 1\}$, surface and interior, to an ellipse.
Under what conditions is the surface of the unit sphere mapped to the surface of the ellipse? (Suggestion: Consider the cases $m \geq n$ and $n > m$ separately. Also, some of the axes of the ellipse may be zero, so it may be convenient to introduce $r \leq \min(m,n)$ such that $\sigma_1 \geq \ldots \geq \sigma_r > 0$.)
I think this is like a SVD question and I found the page here that might be useful: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.3603&rep=rep1&type=pdf
But I still do not quite understand the exact process of mapping the interior and surface of a unit sphere into an ellipse, maybe SVD is doing what I am asking?
Any help will be greatly appreciated.
just take an arbitrary point on the surface of the unit sphere, and apply the transformation to it. There will be a condition that results in the transformed point taking on a form of a k-dimensional ellipse.
-Ivo