I am really sorry if this a very naive question. There are some posts about Singular value decomposition (SVD) that explains SVD from the linear transformation perspective. I sort of understand that. However, I don't understand intuitively what singular values do. According to How can you explain the Singular Value Decomposition to Non-specialists? , a large singular value will indicate that the contribution of the corresponding transformation. I don't really understand what does that contribution means?
According to Understanding singular value decomposition, one geometric interpretation of the singular values of a matrix is the following. Suppose A is a $m×n$ matrix (real-valued, for simplicity). Think of it as a linear transformation $ℝ^n→ℝ^m$ in the usual way. Now take the unit sphere S for $ℝ^n$. Being a linear transformation, A maps S to an ellipsoid in $ℝ^m$. The lengths of the semi-axes of this ellipsoid are precisely the non-zero singular values of A. The zero singular values tell us what the dimension of the ellipsoid is going to be: $n$ minus the number of zero singular values.
If I understand correctly, according to above, the zero singular values are used to determine the dimension of the transformed space. How can the positive singular values be interpreted?