Checkpoint

Eigenvectors and eigenvalues

Twelve questions across the module: finding eigenvectors, what repeated multiplication does, symmetric matrices and singular values. The numbers are new. Work on paper where it helps; the last item is a short piece of code. You pass at 80 percent.

1
Which of these are eigenvectors of $\begin{bmatrix} 3 & 2 \\ 0 & -1 \end{bmatrix}$ ? Select every one.
2
The matrix $\begin{bmatrix} 7 & -2 \\ 4 & 1 \end{bmatrix}$ has two real eigenvalues. Find the eigenvector for the smaller one, scaled so its first entry is 1.
v
3
You learn that $\lambda = 0$ is an eigenvalue of a square matrix $\mathbf{A}$ . What does that tell you?
4
A $3 \times 3$ matrix has trace 4. Two of its eigenvalues are 2 and $-1$ . What is its determinant?
det
5
A matrix $\mathbf{A}$ has eigenvector $(1, 1)$ with eigenvalue 1 and eigenvector $(1, -1)$ with eigenvalue $\tfrac{1}{2}$ . Compute $\mathbf{A}^3(3, 1)$ .
A^3 x
6
What is the spectral radius of $\begin{bmatrix} 0.3 & -0.4 \\ 0.4 & 0.3 \end{bmatrix}$ ?
rho
7
A $2 \times 2$ matrix $\mathbf{W}$ has eigenvalues $0.5$ and $-1.2$ . You multiply a random vector by $\mathbf{W}$ 60 times. What happens?
8
What is the largest singular value of $\begin{bmatrix} 3 & 4 \\ 0 & 0 \end{bmatrix}$ ? Its eigenvalues are 3 and 0.
sigma 1
9
A symmetric $2 \times 2$ matrix has eigenvalues $-4$ and 2. What are its singular values?
10
Centered 2D data has covariance matrix $\begin{bmatrix} 2 & 2 \\ 2 & 5 \end{bmatrix}$ . What is the first principal component, and what fraction of the variance does it capture?
11
You approximate a $512 \times 2048$ weight matrix by its rank-8 truncated SVD, stored as the 8 columns $\sigma_i\mathbf{u}_i$ and the 8 rows $\mathbf{v}_i^\top$ . How many numbers do you store?
numbers
12
Code it
Write gain(A): the largest factor by which the matrix A (any shape) can stretch a vector. Compute it as the square root of the largest eigenvalue of $\mathbf{A}^\top\mathbf{A}$ , using np.linalg.eigvalsh. Do not use np.linalg.svd or np.linalg.norm(A, 2).
⌘↵ runsPython sleeps until you run code

12 still unanswered. Unanswered questions count as wrong.