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:
“Two things in America are astonishing: the changeableness of most human behavior and the strange stability of certain principles. Men are constantly on the move, but the spirit of humanity seems almost unmoved.”
—Alexis de Tocqueville (1805–1859)
“The holy passion of friendship is of so sweet and steady and loyal and enduring a nature that it will last through a whole lifetime, if not asked to lend money.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“To motorists bound to or from the Jersey shore, Perth Amboy consists of five traffic lights that sometimes tie up week-end traffic for miles. While cars creep along or come to a prolonged halt, drivers lean out to discuss with each other this red menace to freedom of the road.”
—For the State of New Jersey, 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)
“We now come to the grand law of the system in which we are placed, as it has been developed by the experience of our race, and that, in one word, is SACRIFICE!”
—Catherine E. Beecher (1800–1878)