Real versus complex nearest normal matrices

Determine whether, for every real matrix A, a real normal matrix nearest to A in the Frobenius norm is also nearest among all complex normal matrices.

Background

The paper studies the problem of approximating a matrix by a nearest normal matrix in the Frobenius norm and develops separate optimization procedures for complex normal matrices and real normal matrices. For a real input matrix, the real algorithm searches only among real normal matrices, whereas the complex formulation permits nonreal normal approximants.

The unresolved issue is whether restricting the admissible approximants to real normal matrices can ever increase the optimal distance compared with allowing all complex normal matrices. The authors report numerical observations suggesting that the real and complex optima may coincide in tested cases, but explicitly note that these computations do not resolve the question because neither algorithm is guaranteed to find a global minimum.