Attainment of the Normal Procrustes Infimum

Determine whether the infimum of the complex Normal Procrustes Problem is always attained by some normal matrix A.

Background

The complex Normal Procrustes Problem minimizes the Frobenius residual ||AX-Y||_F2 over all normal matrices A. The paper notes that the minimizing set may be empty because the normal-matrix constraint is not treated as a compact feasible-set condition in the relevant parametrization. Establishing universal attainment would clarify whether every such problem has an actual optimal normal matrix rather than merely an approximating sequence.

References

Note that this set may be empty; we do not attempt to determine whether the infimum is always attained by some $A \in \mathcal{N}$.

— The Normal Procrustes Problem: A Riemannian Optimization Approach  (2608.19513 - Bierly, 20 Aug 2026) in Section 1, Introduction