let be the Matrix A ...
A is hermitian $A=A^\dagger$
ALL the eigenvalues of A are non zero
then could i solve the linear system $Ax=B$
by using the eigenvectors in this form ?, the solution would be
$ x= \sum_{j}\frac{c_{j}}{\lambda _{j}} V_{j} $
here $ C_j $is the scalar product of the vectors $ C_{j} = (B, V_{j}) $
and $ A V_{j}= \lambda _{j}V_j $ is an eigenvector