Difference equations

Let $ A$ be an $ n$-by-$ n$ matrix, and let $ \mathbf{x}_0$ be an initial vector in $ \mathbb{R}^n$. Then the difference equation

$\displaystyle \mathbf{x}_k = A \mathbf{x}_{k-1}$ (16)

is called a dynamical system, which determines recursively how $ \mathbf{x}_1, \mathbf{x}_2, \ldots$ evolve as time $ k$ passes. If $ A$ is diagonalizable, then the vector $ \mathbf{x}_k$ of the difference equation is obtained in the form of

$\displaystyle \mathbf{x}_{k} = A^k \mathbf{x}_{0}
= P D^k P^{-1} \mathbf{x}_{0}
= c_1 \lambda_1^k \mathbf{v}_{1} + \cdots +
c_n \lambda_n^k \mathbf{v}_{n},$    $ P^{-1} \mathbf{x}_{0} = \begin{bmatrix}c_1  [-0.05in] \vdots  [-0.05in] c_n


Department of Mathematics
Last modified: 2005-10-21