np.linalg.eig() for eigenvalues in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

What is the output of eigenvalues from a 2x2 matrix?

Given the matrix [[4, 2], [1, 3]], what are the eigenvalues computed by np.linalg.eig()?

NumPy

import numpy as np
matrix = np.array([[4, 2], [1, 3]])
eigenvalues, _ = np.linalg.eig(matrix)
print(eigenvalues)

A [5.0, 2.0]

B [3.0, 4.0]

C [6.0, 1.0]

D [4.0, 3.0]

💡 Hint

Recall eigenvalues satisfy the characteristic equation det(A - λI) = 0.

The eigenvalues of the matrix [[4, 2], [1, 3]] are 5 and 2, which satisfy the characteristic polynomial λ² - 7λ + 10 = 0.

❓ data_output

intermediate

Number of eigenvalues returned by np.linalg.eig()

If you apply np.linalg.eig() to a 3x3 matrix, how many eigenvalues will be returned?

NumPy

import numpy as np
matrix = np.eye(3)
eigenvalues, _ = np.linalg.eig(matrix)
print(len(eigenvalues))

A 6

B 1

C 3

D 9

💡 Hint

The number of eigenvalues equals the size of the matrix dimension.

For an n x n matrix, np.linalg.eig() returns n eigenvalues.

🔧 Debug

advanced

Identify the error in eigenvalue calculation code

What error will this code raise?

import numpy as np
matrix = np.array([1, 2, 3, 4])
eigenvalues, eigenvectors = np.linalg.eig(matrix)

NumPy

import numpy as np
matrix = np.array([1, 2, 3, 4])
eigenvalues, eigenvectors = np.linalg.eig(matrix)

A LinAlgError: Last 2 dimensions of the array must be square

B TypeError: 'numpy.ndarray' object is not callable

C ValueError: too many values to unpack (expected 2)

D No error, outputs eigenvalues and eigenvectors

💡 Hint

Check the shape of the input matrix for eig().

np.linalg.eig() requires a square matrix (2D with equal dimensions). A 1D array causes LinAlgError.

🧠 Conceptual

advanced

What does np.linalg.eig() return?

Which of the following best describes the outputs of np.linalg.eig() when applied to a square matrix?

A Only the eigenvectors as a matrix

B A tuple with eigenvalues array and eigenvectors matrix

C Only the eigenvalues as a list

D A single array combining eigenvalues and eigenvectors

💡 Hint

Think about what information is needed to fully describe eigen decomposition.

np.linalg.eig() returns two objects: an array of eigenvalues and a matrix whose columns are the corresponding eigenvectors.

🚀 Application

expert

Find the eigenvalue with the largest magnitude

Given the matrix [[0, -1], [1, 0]], which eigenvalue has the largest absolute value?

NumPy

import numpy as np
matrix = np.array([[0, -1], [1, 0]])
eigenvalues, _ = np.linalg.eig(matrix)
max_abs = max(abs(eigenvalues))
print(max_abs)

A 2.0

B 1.0

C 0.0

D Complex number with magnitude 1.0

💡 Hint

Consider eigenvalues of rotation matrices and their magnitudes.

The matrix represents a 90-degree rotation; eigenvalues are complex numbers with magnitude 1.