Linear Dynamical System - Solution of Linear Dynamical Systems

Solution of Linear Dynamical Systems

If the initial vector is aligned with a right eigenvector of the matrix, the dynamics are simple

$\frac{d}{dt} \mathbf{x}(t) = \mathbf{A} \cdot \mathbf{r}_{k} = \lambda_{k} \mathbf{r}_{k}$

where is the corresponding eigenvalue; the solution of this equation is

$\mathbf{x}(t) = \mathbf{r}_{k} e^{\lambda_{k} t}$

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 .

$\mathbf{x}_{0} = \sum_{k=1}^{N} \left( \mathbf{l}_{k} \cdot \mathbf{x}_{0} \right) \mathbf{r}_{k}$

Therefore, the general solution for is a linear combination of the individual solutions for the right eigenvectors

$\mathbf{x}(t) = \sum_{k=1}^{n} \left( \mathbf{l}_{k} \cdot \mathbf{x}_{0} \right) \mathbf{r}_{k} e^{\lambda_{k} t}$

Similar considerations apply to the discrete mappings.

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