Module 5 · Lesson 4

Eigenvalues & Eigenvectors

⬇ Download notebook ▶ Open in Colab

For a square matrix A, an eigenvector v is a (nonzero) vector whose direction doesn’t change when A is applied to it — only its length scales, by a factor called the eigenvalue λ:

A @ v == λ * v

np.linalg.eig(A) returns (eigenvalues, eigenvectors), where eigenvectors[:, i] is the eigenvector for eigenvalues[i].

For non-square matrices there’s no eigendecomposition, but there is an analogous, even more general tool: the Singular Value Decomposition (np.linalg.svd), which factors any matrix into rotation/scaling components — useful for things like dimensionality reduction, but outside the scope of this lesson’s exercises.

A = np.array([[2.0, 0.0], [0.0, 3.0]])
eigenvalues, eigenvectors = np.linalg.eig(A)
print("eigenvalues: ", eigenvalues)
print("eigenvectors:\n", eigenvectors)

v0 = eigenvectors[:, 0]
lam0 = eigenvalues[0]
print("A @ v0:   ", A @ v0)
print("lam0 * v0:", lam0 * v0)
print("match:", np.allclose(A @ v0, lam0 * v0))

Output:

eigenvalues:  [2. 3.]
eigenvectors:
 [[1. 0.]
 [0. 1.]]
A @ v0:    [2. 0.]
lam0 * v0: [2. 0.]
match: True
4.1

Verify an eigenpair

Write is_eigenpair(A, v, lam, tol=1e-8) that returns True if A @ v is within tol of lam * v (elementwise, via np.allclose), False otherwise.

💡 Hint

np.allclose(A @ v, lam * v, atol=tol).

✅ Solution
def is_eigenpair(A, v, lam, tol=1e-8):
    return np.allclose(A @ v, lam * v, atol=tol)
4.2

Dominant eigenvalue

Write dominant_eigenvalue(A) that returns the eigenvalue of A with the largest absolute value, as a real float (eigenvalues from np.linalg.eig come back as complex numbers even when the imaginary part is zero — use .real after picking the right one, or take abs() for comparison and return the original real part).

💡 Hint

Get eigenvalues from np.linalg.eig(A), use np.argmax(np.abs(eigenvalues)) to find the index of the largest-magnitude one, then take .real.

✅ Solution
def dominant_eigenvalue(A):
    eigenvalues, _ = np.linalg.eig(A)
    idx = np.argmax(np.abs(eigenvalues))
    return eigenvalues[idx].real