Stability and Steady State of The Matrix System
The matrix equation x'(t) = Ax(t) + b with n×1 parameter vector b is stable if and only if all eigenvalues of the matrix A have a negative real part. The steady state x* to which it converges if stable is found by setting x'(t)=0, yielding, assuming A is invertible. Thus the original equation can be written in homogeneous form in terms of deviations from the steady state: . A different way of expressing this (closer to regular usage) is that x* is a particular solution to the in-homogenous equation, and all solutions are in the form, with a solution to the homogenous equation (b=0).
Read more about this topic: Matrix Differential Equation
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