1. Essential Questions
Why Do Eigenvalues And Eigenvectors Capture Long-term Behavior So Effectively? — Quiz
Test your understanding of why do eigenvalues and eigenvectors capture long-term behavior so effectively? with 5 practice questions.
Practice Questions
Question 1
Which description best matches an eigenvector of a matrix $A$?
Question 2
If $A v = 3v$ for a nonzero vector $v$, what is the eigenvalue associated with $v$?
Question 3
Why do eigenvalues and eigenvectors help describe the long-term behavior of repeated matrix multiplication?
Question 4
What usually happens to the component of a vector along an eigenvector when the corresponding eigenvalue satisfies $|\lambda|<1$ and the transformation is applied many times?
Question 5
Suppose $A$ has two eigenvectors, $v_1$ and $v_2$, with eigenvalues $5$ and $\frac{1}{2}$, respectively. If a starting vector has nonzero components along both eigenvectors, what happens after many applications of $A$?