3.6.4 Practice set: eigenvalues

Eigenvalues and eigenvectors

Eigenvalues and eigenvectors (lesson 3.6.1)

0 of 7 correct

1.
$(2, 1)$ is an eigenvector of $\begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix}$ . What is its eigenvalue?
2.
Write the characteristic polynomial of $\begin{pmatrix} 5 & 4 \\ 1 & 2 \end{pmatrix}$ , multiplied out. Type x for $\lambda$
3.
That matrix has the eigenvalue 6. Its eigenvector is $(v_1, 1)$ . Find $v_1$
4.
A $2 \times 2$ matrix has trace 1 and determinant $-6$ . Find its larger eigenvalue
5.
Give the eigenvalue of $\begin{pmatrix} 0 & -4 \\ 1 & 0 \end{pmatrix}$ with a positive imaginary part
6.
$\begin{pmatrix} 3 & 0 \\ 2 & 3 \end{pmatrix}$ has the eigenvalue 3 twice. How many eigenvector lines does it have?
7.
Both diagonal entries of $\begin{pmatrix} 2 & 3 \\ 3 & 2 \end{pmatrix}$ are 2. What is its larger eigenvalue?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Diagonalisation and powers

Diagonalisation and matrix powers (lesson 3.6.2)

0 of 7 correct

1.
$P = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ and $\Lambda = \begin{pmatrix} 5 & 0 \\ 0 & 1 \end{pmatrix}$ . Find the top-left entry of $A = P\Lambda P^{-1}$
2.
$\mathbf{x} = 3\mathbf{v}_1 - \mathbf{v}_2$ , with $A\mathbf{v}_1 = 2\mathbf{v}_1$ and $A\mathbf{v}_2 = 0.5\mathbf{v}_2$ . Then $A^3\mathbf{x} = a\mathbf{v}_1 + b\mathbf{v}_2$ . Find $b$
3.
$A$ has eigenvalues 2 and $-3$ . Find the trace of $A^2$
4.
Each year 30% of town A's people move to B and 10% of B's move to A. What is the steady share of town A?
5.
The unit vector $\mathbf{x} = (0.8, 0.6)$ has $A\mathbf{x} = (4, 3)$ . What eigenvalue does the power method estimate?
6.
$A$ has eigenvalues $-4$ and 3, and $\mathbf{x}$ has parts along both eigenvectors. For large $k$ , $A^k\mathbf{x}$ lies close to the line of which eigenvalue?
7.
Is $\begin{pmatrix} 5 & 1 \\ 0 & 5 \end{pmatrix}$ diagonalisable? Type 1 for yes, 0 for no

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Symmetric matrices

Symmetric matrices and the spectral theorem (lesson 3.6.3)

0 of 7 correct

1.
Find the larger eigenvalue of the symmetric matrix $\begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix}$
2.
Its eigenvector for $\lambda = 5$ is $(2, 1)$ . The other eigenvector is $(1, y)$ . Find $y$
3.
$Q = \begin{pmatrix} 0.6 & -0.8 \\ 0.8 & 0.6 \end{pmatrix}$ is orthogonal. What is the entry in row 1, column 2 of $Q^{-1}$ ?
4.
$A = Q\Lambda Q^\top$ with $\mathbf{q}_1 = (0.6, 0.8)$ , $\mathbf{q}_2 = (-0.8, 0.6)$ and eigenvalues 10 and 5. Find the entry in row 1, column 2 of $A$
5.
eigh returns the eigenvalues $[0.4, 1.1, 7.5]$ of a covariance matrix. What share of the trace is the largest one?
6.
A symmetric matrix has eigenvalues 4 and $-1$ . What is the shortest semi-axis of the ellipse it makes from the unit circle?
7.
eigh reads only the entries on and below the diagonal. What is the larger eigenvalue it returns for $\begin{pmatrix} 1 & 0 \\ 3 & 1 \end{pmatrix}$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

The power method

Write power_method(A, x0, tol=1e-10, max_steps=10_000) that returns (lam, v), as in lesson 3.6.2. lam is the eigenvalue of largest size of the square matrix A, and v a unit eigenvector for it.

Scale x0 to length 1. At each step compute $\mathbf{y} = A\mathbf{x}$ and the estimate $\lambda = \mathbf{x} \cdot \mathbf{y}$ .
Stop when $\lVert \mathbf{y} - \lambda\mathbf{x} \rVert$ is below tol, which says $A\mathbf{x} = \lambda\mathbf{x}$ to within tol. Otherwise set $\mathbf{x} = \mathbf{y} / \lVert \mathbf{y} \rVert$ and go on.
Return v with length 1 and its entry of largest size positive, so that the answer does not depend on the sign.
Raise ValueError if A is not square, or x0 is zero or has the wrong length. Raise RuntimeError after max_steps steps without stopping.

The tests include lesson 3.6.3's $3 \times 3$ matrix, a dominant eigenvalue of $-6$ , a start that A sends to $\mathbf{0}$ and the quarter turn. A version that stops when $\mathbf{x}$ stops changing never stops for the $-6$ : work out why before you run it.

See the tests

import numpy as np


def test_symmetric():
    """S = [[2, 1], [1, 2]] from (1, 0): λ = 3 and v = (1, 1)/√2"""
    lam, v = power_method([[2, 1], [1, 2]], [1, 0])
    expect_close(lam, 3.0, what="eigenvalue", rel=1e-9)
    expect_close(v, np.array([1, 1]) / np.sqrt(2), what="eigenvector", abs_tol=1e-6)


def test_not_symmetric():
    """Lesson 3.6.2's A = [[1, 2], [3, 2]] from (1, 0): λ = 4 and v = (2, 3)/√13"""
    lam, v = power_method([[1, 2], [3, 2]], [1, 0])
    expect_close(lam, 4.0, what="eigenvalue", rel=1e-9)
    expect_close(v, np.array([2, 3]) / np.sqrt(13), what="eigenvector", abs_tol=1e-6)


def test_steady_state():
    """The two towns, M = [[0.9, 0.2], [0.1, 0.8]]: λ = 1 and v = (2, 1)/√5"""
    lam, v = power_method([[0.9, 0.2], [0.1, 0.8]], [1, 0])
    expect_close(lam, 1.0, what="eigenvalue", rel=1e-9)
    expect_close(v, np.array([2, 1]) / np.sqrt(5), what="eigenvector", abs_tol=1e-6)


def test_three_by_three():
    """Lesson 3.6.3's B from (1, 0, 0): λ = 6 and v = (−1, 2, −1)/√6, its largest entry positive"""
    B = [[3, -1, 1], [-1, 5, -1], [1, -1, 3]]
    lam, v = power_method(B, [1, 0, 0])
    expect_close(lam, 6.0, what="eigenvalue", rel=1e-9)
    expect_close(v, np.array([-1, 2, -1]) / np.sqrt(6), what="eigenvector", abs_tol=1e-6)


def test_unit_length():
    """The eigenvector returned has length 1, whatever the length of x0"""
    _, v = power_method([[1, 2], [3, 2]], [100, 0])
    expect_close(float(np.linalg.norm(v)), 1.0, what="length of v", rel=1e-12)


def test_negative_eigenvalue():
    """[[−5, 2], [2, −2]] has λ = −6: x flips at every step, yet λ = −6 and v = (2, −1)/√5"""
    lam, v = power_method([[-5, 2], [2, -2]], [1, 0])
    expect_close(lam, -6.0, what="eigenvalue", rel=1e-9)
    expect_close(v, np.array([2, -1]) / np.sqrt(5), what="eigenvector", abs_tol=1e-6)


def test_slow():
    """[[1, 0.01], [0.01, 1]], with eigenvalues 1.01 and 0.99, still converges: λ = 1.01"""
    lam, _ = power_method([[1, 0.01], [0.01, 1]], [1, 0])
    expect_close(lam, 1.01, what="eigenvalue", rel=1e-9)


def test_zero_eigenvalue():
    """Z = [[1, 2], [0.5, 1]] sends x0 = (2, −1) to 0: λ = 0 and v = (2, −1)/√5"""
    lam, v = power_method([[1, 2], [0.5, 1]], [2, -1])
    expect_close(lam, 0.0, what="eigenvalue", abs_tol=1e-12)
    expect_close(v, np.array([2, -1]) / np.sqrt(5), what="eigenvector", abs_tol=1e-9)


def test_rotation():
    """A quarter turn keeps no direction: RuntimeError after max_steps"""
    try:
        power_method([[0, -1], [1, 0]], [1, 0], max_steps=500)
    except RuntimeError:
        return
    raise AssertionError("expected RuntimeError for the quarter turn, which has no real eigenvector")


def test_max_steps():
    """Five steps are not enough for S from (1, 0) at tol 1e-10: RuntimeError"""
    try:
        power_method([[2, 1], [1, 2]], [1, 0], max_steps=5)
    except RuntimeError:
        return
    raise AssertionError("expected RuntimeError after 5 steps")


def test_bad_inputs():
    """A 2 × 3 matrix, a zero start, or a start of the wrong length: ValueError"""
    for A, x0, what in [([[1, 2, 3], [4, 5, 6]], [1, 0, 0], "a 2 × 3 matrix"),
                        ([[2, 1], [1, 2]], [0, 0], "x0 = (0, 0)"),
                        ([[2, 1], [1, 2]], [1, 0, 0], "x0 with 3 entries for a 2 × 2 matrix")]:
        try:
            power_method(A, x0)
        except ValueError:
            continue
        raise AssertionError(f"expected ValueError for {what}")

Eigenvalues and eigenvectors

Diagonalisation and powers

Symmetric matrices

Coding exercise

Free further reading