Complex Eigenvalues and Dynamics

A 2 × 2 matrix with complex eigenvalues can be expressed in the form of a rotation matrix combined with a scaling factor, which allows it to describe transformations involving both size and direction.

The scaling component is given by the magnitude of the complex eigenvalue, while the rotation component is determined by its argument, representing the angle of rotation.

The origin is classified based on the eigenvalues of the system; it is an attractor if all eigenvalues have magnitudes less than 1, a repeller if they exceed 1, and a saddle point if at least one is greater than 1 and another is less than 1.

<p>The long-term behavior of trajectories is influenced by the eigenvalues of the transition matrix, which dictate whether trajectories converge to fixed points or diverge away.</p> Signup and view all the answers

<p>The trajectory can be represented as $ extbf{x}_k = A^k extbf{x}_0$, where $A$ is the transition matrix, $k$ is the number of iterations, and $ extbf{x}_0$ is the initial state, expressed in terms of the eigenvalues and eigenvectors of $A$.</p> Signup and view all the answers

A 2x2 matrix with complex eigenvalues is similar to a matrix representing a scaling and rotation.

A system where the state changes over time in discrete steps.

Terms used to describe the behavior of trajectories in a discrete dynamical system near a fixed point (origin).

A square matrix where each entry represents the probability of moving from one state to another.

A vector representing the long-term distribution of probabilities in a stochastic system.