THE REAL AND THE SYMMETRICAL NONNEGATIVE INVERSE EIGENVALUE PROBLEMS ARE DIFFERENT

We show that there exist real numbers lambda(1),lambda(2),...,lambda(n ) that occur as the eigenvalues of an entry-wise nonnegative n-by-n ma trix but do not occur as the eigenvalues of a symmetric nonnegative n- by-n matrix. This solves a Problem posed by Boyle and Handelman, Hersh kowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by appending O 's to given spectral data is refined.