In this paper, we propose a change of coordinates that brings the state matrix of an autonomous linear system into a modified Jordan Block form. This change of coordinates and the obtained modified Jordan Block form, allows us to obtain exact values of the scaling factor and the convergence rate of the exponential stability bound for linear systems. The analysis is then applied to obtain sufficient conditions for the stability of semilinear systems with Lipshitz nonlinearity. Numerical examples are also provided to illustrate the effectiveness of the proposed approach.