In the context of the Lyapunov equation and Lyapunov functions, for linear systems, one can try to find a Lyapunov function in the quadratic form:
In this case, V(x) is greater than zero if P is greater than zero, or in other words, if P is a symmetric and positive definite matrix.
By positive definite, we mean that all eigenvalues of the matrix are positive. Another way to examine positive-definiteness is the requirement that the determinants of all upper-left submatrices are positive.