Unit 5 test: Eigenvalues and Eigenvectors

Let $A = \begin{bmatrix} 3 & 1 \\ 4 & 3 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$. The vector $\mathbf{v}$ is an eigenvector of $A$. Find its eigenvalue.

Find all eigenvalues of $A = \begin{bmatrix} 2 & 5 & -1 \\ 0 & -3 & 4 \\ 0 & 0 & 7 \end{bmatrix}$.

Let $A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 1 & 2 & 5 \end{bmatrix}$. Find the dimension of the eigenspace of $A$ for $\lambda = 3$.

Let $A$ be an $n \times n$ matrix. Which statement is equivalent to "$0$ is an eigenvalue of $A$"?

Find the eigenvalues of $A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}$.

Find the eigenvalues of $A = \begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 4 \end{bmatrix}$.

A $3 \times 3$ matrix $A$ has eigenvalues $2$, $-1$ and $4$. Find the trace of $A^2$.

Let $A = PDP^{-1}$ with $P = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $D = \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}$. Find the $(1, 2)$ entry of $A^3$.

Which matrix is not diagonalizable?

Let $A = \begin{bmatrix} 4 & -1 & 2 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$. Write $A = PDP^{-1}$ with $D = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$ and first column of $P$ equal to $(1, 0, 0)$. The second column of $P$ must be an eigenvector for $\lambda = 3$ whose first entry is $1$. Find it.

Find the eigenvalues of $A = \begin{bmatrix} 1 & -2 \\ 1 & 3 \end{bmatrix}$.

The matrix $C = \begin{bmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{bmatrix}$ acts on $\mathbb{R}^2$ as a rotation followed by a scaling. Find the rotation angle in degrees, between $0$ and $360$.

Find the steady-state vector of the stochastic matrix $P = \begin{bmatrix} 0.7 & 0.2 \\ 0.3 & 0.8 \end{bmatrix}$.

A Markov chain has transition matrix $P = \begin{bmatrix} 0.7 & 0.2 \\ 0.3 & 0.8 \end{bmatrix}$ and starts in state 2, so $\mathbf{x}_0 = (0, 1)$. Find the probability that it is in state 1 after two steps.

Let $P$ be a regular stochastic matrix with steady-state vector $\mathbf{q}$. Which statement is true?