  Eigenvalue-preserving-but-not-completely-eigenvalue-preserving (EnCE) maps
were previously introduced for the purpose of detection and quantification of
nonclassical correlation, employing the paradigm where nonvanishing quantum
discord implies the existence of nonclassical correlation. It is known that
only the matrix transposition is nontrivial among Hermiticity-preserving (HP)
linear EnCE maps when we use the changes in the eigenvalues of a density matrix
due to a partial map for the purpose. In this paper, we prove that this is true
even among not-necessarily HP (nnHP) linear EnCE maps. The proof utilizes a
conventional theorem on linear preservers. This result imposes a strong
limitation on the linear maps and promotes the importance of nonlinear maps.
