Summary: Let \(s_1, \ldots, s_n\) be arbitrary complex scalars. It is required to construct an \(n \times n\) normal matrix \(A\) such that \(s_1\) is an eigenvalue of the leading principal submatrix \(A_i\), \(i = 1, 2, \ldots, n\). It is shown that, along with the obvious diagonal solution \(\text{diag} (s_1, \ldots, s_n)\), this problem always admits a much more interesting nondiagonal solution \(A\). As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix \(A_i\) is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices \(A_i\) and \(A_{i + 1}\).