12. Diagonalization and Dynamical Systems

Long-term Behavior — Quiz

Test your understanding of long-term behavior with 5 practice questions.

Practice Questions

Question 1

A diagonalizable matrix $A$ has eigenvalues $5$ and $2$. If an initial vector has a nonzero component in each eigendirection, which eigenvalue determines the long-term behavior of $A^k x_0$?

Question 2

For the system $x_{k+1}=Ax_k$, suppose every eigenvalue $$ of $A$ satisfies $||<1$. What happens to $x_k$ as $k\to\infty$?

Question 3

A diagonalizable matrix has a dominant eigenvalue of $-3$. If an initial vector has a nonzero component in that eigendirection, what happens to that component over time?

Question 4

For the system $x_{k+1}=Ax_k$, suppose $v$ is an eigenvector with eigenvalue $1$. What happens to the component of a solution in the direction of $v$ after repeated steps?

Question 5

A diagonalizable matrix has eigenvalues $2$ and $1/2$. Which eigenvalue controls the long-term behavior of $A^k x_0$ when $x_0$ has components in both eigendirections?