  Fillmore Theorem says that if A is an nxn complex non-scalar matrix and
{\gamma}_1,...,{\gamma}_{n} are complex numbers with
{\gamma}_1+...+{\gamma}_{n}=trA, then there exists a matrix B similar to A with
diagonal entries {\gamma}_1,...,{\gamma}_{n}. Borobia simplifies this result
and extends it to matrices with integer entries. Fillmore and Borobia do not
consider the nonnegativity hypothesis. Here, we introduce a different and very
simple way to compute the matrix B similar to A with diagonal
{\gamma}_1,...,{\gamma}_{n}. Moreover, we consider the nonnegativity hypothesis
and we show that for a list {\Lambda}={{\lambda}_1,...,{\lambda}_{n}} of
complex numbers of Suleimanova or \v{S}migoc type, and a given list
{\Gamma}={{\gamma}_1,...,{\gamma}_{n}} of nonnegative real numbers, the
remarkably simple condition
{\gamma}_1+...+{\gamma}_{n}={\lambda}_1+...+{\lambda}_{n} is necessary and
sufficient for the existence of a nonnegative matrix with spectrum {\Lambda}
and diagonal entries {\Gamma}. This surprising simple result improves a
condition recently given by Ellard and \v{S}migoc in arXiv:.1702.02650v1.
