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)
“I should like a lover to think of the things that I think about. It is all very well being steady when you have got babies of your own; but that should be after ever so long. I should like to keep my lover as a lover for two years. And all that time he should like to dance with me, and to hear music, and to go about just when I would like to go.”
—Anthony Trollope (1815–1882)
“Wags try to invent new stories to tell about the legislature, and end by telling the old one about the senator who explained his unaccustomed possession of a large roll of bills by saying that someone pushed it over the transom while he slept. The expression “It came over the transom,” to explain any unusual good fortune, is part of local folklore.”
—For the State of Montana, U.S. public relief program (1935-1943)
“In all cultures, the family imprints its members with selfhood. Human experience of identity has two elements; a sense of belonging and a sense of being separate. The laboratory in which these ingredients are mixed and dispensed is the family, the matrix of identity.”
—Salvador Minuchin (20th century)
“New York is more now than the sum of its people and buildings. It makes sense only as a mechanical intelligence, a transporter system for the daily absorbing and nightly redeploying of the human multitudes whose services it requires.”
—Peter Conrad (b. 1948)