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 quotation contributes something to the stability or enlargement of the language.”
—Samuel Johnson (1709–1784)
“Isn’t it odd that networks accept billions of dollars from advertisers to teach people to use products and then proclaim that children aren’t learning about violence from their steady diet of it on television!”
—Toni Liebman (20th century)
“In days gone by, we were afraid of dying in dishonor or a state of sin. Nowadays, we are afraid of dying fools. Now the fact is that there is no Extreme Unction to absolve us of foolishness. We endure it here on earth as subjective eternity.”
—Jean Baudrillard (b. 1929)
““The matrix is God?”
“In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
“If it has limits, it isn’t omnipotent.”
“Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””
—William Gibson (b. 1948)
“Television is an excellent system when one has nothing to lose, as is the case with a nomadic and rootless country like the United States, but in Europe the affect of television is that of a bulldozer which reduces culture to the lowest possible denominator.”
—Marc Fumaroli (b. 1932)