General Normal Targeting Problem

Solve the general Normal Targeting Problem by characterizing when given matrices X and Y admit a normal matrix A satisfying AX=Y.

Background

The Normal Targeting Problem asks for necessary and sufficient conditions under which AX=Y for a normal matrix A. The paper reports partial results when the spectrum of A is restricted to two prescribed values, but states that the unrestricted problem remains unresolved. A general solution would provide exact solvability criteria complementary to the paper’s numerical least-squares approach.

References

The case in which $A$ is normal was found to be particularly difficult. Indeed, smaller results were found, such as when the spectrum of a normal $A$ is constrained to lie in ${\lambda,\mu},$ where $\lambda,\mu \in C$, but the general Normal Targeting Problem remains unsolved.

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