Solution of Linear Dynamical Systems
If the initial vector is aligned with a right eigenvector of the matrix, the dynamics are simple
where is the corresponding eigenvalue; the solution of this equation is
as may be confirmed by substitution.
If is diagonalizable, then any vector in an -dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted ) of the matrix .
Therefore, the general solution for is a linear combination of the individual solutions for the right eigenvectors
Similar considerations apply to the discrete mappings.
Read more about this topic: Linear Dynamical System
Famous quotes containing the words solution of, solution and/or systems:
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (18901960)
“To the questions of the officiously meddling police Falter replied absently and tersely; but, when he finally grew tired of this pestering, he pointed out that, having accidentally solved the riddle of the universe, he had yielded to artful exhortation and shared that solution with his inquisitive interlocutor, whereupon the latter had died of astonishment.”
—Vladimir Nabokov (18991977)
“People stress the violence. Thats the smallest part of it. Football is brutal only from a distance. In the middle of it theres a calm, a tranquility. The players accept pain. Theres a sense of order even at the end of a running play with bodies stewn everywhere. When the systems interlock, theres a satisfaction to the game that cant be duplicated. Theres a harmony.”
—Don Delillo (b. 1926)