Summary: Given three \(n\)-tuples \(\{\lambda_i\}_{i=1}^n\) , \(\{\mu_i\}_{i=1}^n\), \(\{\nu_i\}_{i=1}^n\) of complex numbers, we introduce the problem of when there exists a pair of normal matrices \(A\) and \(B\) such that \(\sigma (A) = \{\lambda_i\}_{i=1}^n\), \(\sigma (B) = \{\mu_i\}_{i=1}^n\), and \(\sigma (A + B) = \{\nu_i\}_{i=1}^n\), where \(\sigma (\cdot )\) denote the spectrum. In the case when \(\lambda_k = 0\), \(k = 2,\dots , n\), we provide necessary and sufficient conditions for the existence of \(A\) and \(B\). In addition, we show that the solution pair \((A, B)\) is unique up to unitary similarity. The necessary and sufficient conditions reduce to the classical A. Horn inequalities when the \(n\)-tuples are real.