  In this paper, the concept of generalized spectral function is introduced for
finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex
entries. The structure of the generalized spectral function is described in
terms of spectral data consisting of the eigenvalues and normalizing numbers of
the matrix. The inverse problems from generalized spectral function as well as
from spectral data are investigated. In this way, a procedure for construction
of complex tridiagonal matrices having real eigenvalues is obtained.
