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The almost-sure asymptotic stability of finite order linear stochastic systems subjected to non-white parametric excitations can be analyzed via the second method of Lyapunov with a quadratic function. In a previous work, it is reported that this approach produces an inequality condition among system parameters and some free optimization variables for a.s. asymptotic stability. However, in this study, utilizing the properties of the Kronecker Products it is shown that this inequality condition can only be used provided that certain other conditions are satisfied. These necessary conditions for the fourth order systems are explicitly obtained using the symbolic computation system Mathematica. Even though these conditions for higher order systems (N>4) are not available in an explicit form due to absence of formulae for roots of a polynomial whose degree is higher than four, a method proposed here provides a formulation for numerical and/or hybrid calculations (symbolic and numerical variables together). The need for this formulation stems from the fact that a pure numerical approach may not be efficient, if possible at all. In addition to stochastic systems, the inequality can be used to analyze the stability of linear time-varying deterministic systems due to the way the second method of Lyapunov is used. This is exhibited with the stability analysis of the damped Mathieu equation.
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