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
Famous quotes containing the words stability, steady, state, matrix and/or system:
“Every nation ... whose affairs betray a want of wisdom and stability may calculate on every loss which can be sustained from the more systematic policy of its wiser neighbors.”
—James Madison (1751–1836)
“What I admire most in any man is a serene spirit, a steady freedom from moral indignation, and all-embracing tolerance—in brief, what is commonly called sportsmanship.”
—H.L. (Henry Lewis)
“It should be noted that when he seizes a state the new ruler ought to determine all the injuries that he will need to inflict. He should inflict them once and for all, and not have to renew them every day.”
—Niccolò Machiavelli (1469–1527)
“As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”
—Margaret Atwood (b. 1939)
“As long as learning is connected with earning, as long as certain jobs can only be reached through exams, so long must we take this examination system seriously. If another ladder to employment was contrived, much so-called education would disappear, and no one would be a penny the stupider.”
—E.M. (Edward Morgan)