Nonuniqueness of induced-norm-minimizing right inverses

Every right inverse of a full-row-rank $A\in\mathbb C^{m\times n}$ has the form $A^\dagger+N$ with $AN=0$, and $\|(A^\dagger+N)z\|_2^2=\|A^\dagger z\|_2^2+\|Nz\|_2^2$, so the Moore–Penrose inverse minimizes the induced $\ell^p\to\ell^2$ norm among right inverses for every $p$. A single $2\times3$ full-row-rank matrix nevertheless has, for every $2<p<\infty$ and over both fields, multiple right inverses minimizing this norm, with common norm $2^{1/2-1/p}$. The complete minimizer set is a disk over the complex field and an interval over the reals. A complex higher-dimensional construction, built on Fourier vectors whose entries have equal modulus, gives a ball of minimizers for every $2\le m<n$. The smallest matrix is attributed to Dokmanić and Gribonval's spectral-norm example; its construction is not claimed as new.