In this paper, we discuss inverse eigenvalue problems
for singular Hermitian matrices. In particular, we investigate how to construct
nún singular Hermitian matrices of rank 2 and 3 from a given prescribed spectral
data. It is found that given the spectrum and the multipliers ki where i = 1,2,3,...,n-r, the inverse eigenvalue problem for n x
n singular Hermitian matrices of rank r is solvable. Numerical
examples are presented in each case.