Let $T: V \rightarrow W$ be an operator between to inner product spaces. Then singular values $s_1 \leq s_2 .... \leq s_n$ of $T$ are square roots of eigenvalues of $T^*T$ where $T^*$ is the conjugate with respect to given inner products.
Q1- Is there a special name for these eigenvectors which give the singular values, like principal directions or something. This I ask because the directions which give respectively the largest and smallest singular values are (one of) the most expanding and contracting directions. In some cases also the other directions might have some importance mainly because they are mapped to orthogonal directions in side W. So I am wondering if they have a special name in literature.
Q2- In the case dim($V$)=dim($W$)=2, the product of the singular values $s_1s_2$ gives the determinant or the volume growth of $T$. In higher dimensions I have seen cases where still the product of $s_{n-1}s_n$ having certain properties had important results. I am wondering if in literature there is any name or geometrical meaning attached to the product of two largest singular values of a map (apart from the fact that it will be one of the largest singular values of $T \wedge T : V\wedge V \rightarrow W\wedge W$)
Q3- Are there any tools (apart from Courant-Fischer minmax and Cauchy interlacing) that allows one to analyze singular values in finite dimensions.
Thanks
Q1: These are right singular vectors, see Singular value decomposition
Q2: The geometric meaning of the product of two largest singular values is the maximal amount of area increase under the map. That is, if you have a 2-dimensional plane (or surface) transformed by $T$, the area will increase by at most $\sigma_{n-1}\sigma_n$.
Q3: I can't really say, without knowing what is to be analyzed. Is it a computational problem of finding SVD of given matrix? There are a few algorithms for that. Do you want to get an estimate for singular values based on some information about the matrix? It will matter what the information is. Generally, I think minimax is the most convenient theoretical tool.