The imbeddability for hermitian and normal matrices

Let w(A) be the numerical range of a matrix A∈Cn×n and a set of points μ1,…,μn-k∈w(A) that define the spectrum σ(B) of a matrix B∈C(n-k)×(n-k). The problem of imbedding concerns the existence and construction of an isometry V∈Cn×(n-k) such that B=V∗AV and is undertaken in this paper. We initially deal with hermitian matrices, for which it is well known that the necessary and sufficient condition for the imbeddability of matrix B in A is their eigenvalue interlacing, and here we present a formulation for the isometry V, when k⩾1. Moreover, concerning normal matrices, a criterion for imbeddability is established in terms of the real and imaginary parts of their eigenvalues.