  We consider a generalization of the full symmetric Toda hierarchy where the
matrix $\tilde {L}$ of the Lax pair is given by $\tilde {L}=LS$, with a full
symmetric matrix $L$ and a nondegenerate diagonal matrix $S$. The key feature
of the hierarchy is that the inverse scattering data includes a class of
noncompact groups of matrices, such as $O(p,q)$. We give an explicit formula
for the solution to the initial value problem of this hierarchy. The formula is
obtained by generalizing the orthogonalization procedure of Szeg\"{o}, or the
QR factorization method of Symes. The behaviors of the solutions are also
studied. Generically, there are two types of solutions, having either sorting
property or blowing up to infinity in finite time. The $\tau$-function
structure for the tridiagonal hierarchy is also studied.
