Given a function f : $\mathbb{R}^n\to\mathbb{R}^m$, which is known to be Lipschitz continuous, can we say anything about the Lipschitz continuity of it's inverse function (in this case, the pseudoinverse matrix) ?
[Math] Lipschitz continuity of inverse
continuityinverselinear algebramatrices
Best Answer
Every linear map $f:\mathbb R^n\to\mathbb R^m$ is represented by a $(m\times n)$ matrix, say $A$. The operator norm of $A$ is equal to the Lipschitz constant of $f$, directly from definitions.
Same holds for the linear map defined by the pseudoinverse matrix $A^\dagger$. The map is Lipschitz, with the Lipschitz constant $\|A^\dagger \|$.
You should be aware that the linear transformation given by the pseudoinverse matrix $A^\dagger$ need not be an inverse map in set-theoretical sense; i.e., $AA^\dagger$ and $A^\dagger A$ are generally not identity matrices.