This technical note is concerned with stability analysis and stabilization of Ito stochastic systems with Markovian switching. A couple of eigenvalue sets for some positive operator associated with the stochastic system under study are defined to characterize its stability in the mean square sense. Properties for these eigenvalue sets are established based on which we show that the spectral abscissa of these eigenvalues sets are the same and thus these eigenvalue sets are equivalent in the sense of characterizing the stability of the system. Also, it is shown that the guaranteed convergence rate of the Markovian jump Ito stochastic systems can be determined by some eigenvalue set. Finally, a linear matrix inequality based approach is proposed to design controllers such that the closed-loop system has guaranteed convergence rate. Some numerical examples are carried out to illustrate the effectiveness of the proposed approach. The research in this technical note opens several perspectives for future work stated as some open problems.