Interaction between Hermitian and normal imbeddings

For several applications, it is highly desirable to understand how the eigenvalues of an imbeddable matrix V∗AV, where V∈Cn×k is an isometry, are distributed throughout the numerical range of A∈Cn×n. There has been extensive study for A Hermitian, while a geometric description for the eigenvalues of imbeddings in non-Hermitian matrices remains a challenging problem. Toward this direction, a subspace is introduced, wherein all n×n complex diagonal matrices for which a given isometry V∈Cn×k generates diagonal imbeddings are defined. In particular, conditions upon which a real diagonal matrix may be imbeddable in some normal are obtained, including an application for higher rank numerical ranges. Finally, a procedure determining whether two given sets of complex numbers may be realized as spectra of a pair of imbeddable normal matrices is established.