A family of sharp, arbitrarily tight upper and lower matrix bounds far solutions of the discrete algebraic Lyapunov are presented. The lower bounds are tighter than previously. known ones. Unlike the majority of previously known upper bounds? those derived here have no restrictions on their applicability. Upper and lower bounds for individual eigenvalues and for the trace of the solution are found using the new matrix bounds. Sharp trace hounds not derivable from the matrix bounds are also presented.