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.

Practice

1. What does the function np.linalg.eig() return when applied to a square matrix?

easy

A. The determinant and inverse of the matrix

B. An array of eigenvalues and a matrix of eigenvectors

C. The transpose and trace of the matrix

D. The sum and product of matrix elements

Solution

1. Step 1: Understand the purpose of np.linalg.eig()

   This function is designed to find eigenvalues and eigenvectors of a square matrix.

2. Step 2: Recall the output format

   It returns two objects: one array with eigenvalues and one matrix with eigenvectors as columns.

3. Final Answer:

   An array of eigenvalues and a matrix of eigenvectors -> Option B

4. Quick Check:

   Eigenvalues and eigenvectors [OK]

Hint: Remember: eig() returns eigenvalues and eigenvectors [OK]

Common Mistakes:

• Confusing eigenvalues with determinant
• Expecting only one output instead of two
• Mixing eigenvectors with matrix transpose

2. Which of the following is the correct syntax to compute eigenvalues and eigenvectors of matrix A using NumPy?

easy

A. eigenvalues, eigenvectors = np.linalg.eig(A)

B. eigenvalues = np.linalg.eigvals(A)

C. eigenvectors, eigenvalues = np.linalg.eig(A)

D. eigenvalues, eigenvectors = np.eig.linalg(A)

Solution

1. Step 1: Recall the correct function and output order

   The function np.linalg.eig() returns eigenvalues first, then eigenvectors.

2. Step 2: Check syntax correctness

   eigenvalues, eigenvectors = np.linalg.eig(A) correctly assigns eigenvalues and eigenvectors in order from np.linalg.eig(A).

3. Final Answer:

   eigenvalues, eigenvectors = np.linalg.eig(A) -> Option A

4. Quick Check:

   Correct function and order [OK]

Hint: eig() returns (values, vectors) in that order [OK]

Common Mistakes:

• Swapping eigenvalues and eigenvectors in assignment
• Using wrong function like np.linalg.eigvals() for both outputs
• Incorrect module or function name

3. Given the matrix A = np.array([[2, 0], [0, 3]]), what will be the output of np.linalg.eig(A)[0]?

medium

A. [3. 2.]

B. [0. 0.]

C. [2. 3.]

D. [5. 0.]

Solution

1. Step 1: Identify eigenvalues of a diagonal matrix

   For a diagonal matrix, eigenvalues are the diagonal elements: 2 and 3.

2. Step 2: Check the output of np.linalg.eig(A)[0]

   This returns the eigenvalues array, which will be [2. 3.].

3. Final Answer:

   [2. 3.] -> Option C

4. Quick Check:

   Diagonal elements = eigenvalues [OK]

Hint: Diagonal matrix eigenvalues = diagonal elements [OK]

Common Mistakes:

• Confusing eigenvalues order
• Expecting eigenvectors instead of eigenvalues
• Misreading matrix elements

4. What is wrong with this code snippet?

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

medium

A. np.linalg.eigvals() returns only eigenvalues, not eigenvectors

B. Matrix A is not square

C. np.linalg.eigvals() requires two arguments

D. The import statement is incorrect

Solution

1. Step 1: Understand the function np.linalg.eigvals()

   This function returns only eigenvalues, not eigenvectors.

2. Step 2: Check the assignment in the code

   The code tries to unpack two values, but eigvals() returns only one, causing an error.

3. Final Answer:

   np.linalg.eigvals() returns only eigenvalues, not eigenvectors -> Option A

4. Quick Check:

   eigvals() returns one output [OK]

Hint: eigvals() returns only eigenvalues, not vectors [OK]

Common Mistakes:

• Expecting two outputs from eigvals()
• Thinking matrix must be non-square
• Misunderstanding import syntax

5. You have a matrix B = np.array([[1, 0], [0, -3]]). You want to find the eigenvalue with the largest magnitude. Which code snippet correctly finds it?

hard

A. vals, vecs = np.linalg.eig(B)
largest = max(vecs)

B. vals = np.linalg.eigvals(B)
largest = max(vals)

C. vals = np.linalg.eigvals(B)
largest = vals[np.argmax(vals)]

D. vals, vecs = np.linalg.eig(B)
largest = vals[np.argmax(np.abs(vals))]

Solution

1. Step 1: Compute eigenvalues and eigenvectors

   Use np.linalg.eig(B) to get both eigenvalues and eigenvectors.

2. Step 2: Find eigenvalue with largest magnitude

   Use np.abs(vals) to get absolute values, then np.argmax() to find index of largest magnitude eigenvalue.

3. Step 3: Select eigenvalue at that index

   Indexing vals with that index gives the eigenvalue with largest magnitude.

4. Final Answer:

   vals, vecs = np.linalg.eig(B) largest = vals[np.argmax(np.abs(vals))] -> Option D

5. Quick Check:

   Use abs and argmax on eigenvalues [OK]

Hint: Use np.abs() and np.argmax() on eigenvalues to find largest [OK]

Common Mistakes:

• Using max() directly without abs()
• Trying to find max of eigenvectors
• Using wrong function for eigenvalues