Generalization of Mirsky's theorem on diagonals and eigenvalues of   matrices
  Mirsky proved that, for the existence of a complex matrix with given eigenvalues and diagonal entries, the obvious necessary condition is also sufficient. We generalize this theorem to matrices over any field and provide a short proof. Moreover, we show that there is a unique companion-matrix-type solution for this problem. 
Dragomir Z. Djokovic
math.RA