I am trying to visualize in MATLAB the relationship between the singular value decomposition (SVD) of a matrix of points. To simplify the problem, I am working in 2D and I am considering an ellipse whose $n$ points $\mathbf{p}_i$ are column vectors of the matrix $\mathbf{M}$:
$$
\mathbf{p}_i = \begin{bmatrix}
x_i \\ y_i
\end{bmatrix}, \quad i = 1, 2, \ldots, n
$$
To begin with, I considered an ellipse with its major axis parallel to the $x$-axis:
Then, I computed the SVD, which allows me to decompose $\mathbf{M}$ as: $$ \mathbf{M} = \mathbf{U}_m \mathbf{S}_m \mathbf{V}_m $$ From the columns of $\mathbf{U}_m$, I derived the directions of the principal axes of the ellipse, and I plotted them:
At this point, I looked at the content of the matrix $\mathbf{S}_m$, and I expected to find the length of the ellipse's axes. However, this is what the matrix contains (I disregarded the part of the matrix containing only zeros on its diagonal): $$ \mathbf{S}_m = \begin{bmatrix} 11.6336 & 0 \\ 0 & 8.1649 \end{bmatrix} $$
As a result, here is my question:
- how do I relate the principal values contained in $\mathbf{S}_m$ to the actual length of the axes?