Smallest singular value interpretation as inverse spectral norm?

Question

The largest singular value of a finite dimensional matrix is its spectral norm (L2 operator norm). In other words, it is the maximum scaling that A does to any vector. Does the smallest singular value have any interesting properties? For instance, if the singular values are all lower bounded by 0, then could $\sigma_{min}$ it be interpreted as a kind of "inverse" spectral norm; i.e. the minimum the matrix must scale any vector?

Answer 1

Yes: $\sigma_{min}^2$ is the least eigenvalue of $A^* A$, which is the minimum of $x^* A^* A x = \|A x\|^2$ for $x$ with $\|x\| = 1$.