Question

It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:

$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.

Suppose we define a different function of matrices this way:

$f(A)=\inf_{x \neq 0}{\frac{||Ax||}{||x||}}$.

Has $f(\cdot)$ been studied before? Does it have a standard name?

Answer 1

By the Courant-Fischer min-max theorem, if $A$ is Hermitian, then $f(A) = \lambda_n(A)$, the smallest eigenvalue of $A$.

Answer 2

$|||A|||$ is the largest s-number (modulus of gen. eigenvalue). $f(A)$ is the smallest s-number. It is 0 if $A$ is not injective.