If we have a set of vectors $\{ \vec{v}_i\}$, then the corresponding Grammian matrix has entries which are inner products of the different vectors in the set. The Grammian matrix is always Hermitian, and positive semi-definite. I have two questions about the eigenvalues, which are possibly a bit too open-ended:
- A zero eigenvalue signifies a linear dependence relationship between the vectors. Is there an interpretation about non-zero eigenvalues? In particular, is there a notion of "larger" independence? If yes, is this somehow connected to the magnitude of the eigenvalues (assuming that the vectors are normalized to begin with)?
- Is it possible for the Grammian matrix to have degenerate eigenvalues? If yes, does the degeneracy have an interpretation in terms of the original vectors? Thank you for the help!
