np.linalg.eig() for eigenvalues in NumPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to compute eigenvalues of matrix A.

NumPy

import numpy as np
A = np.array([[2, 0], [0, 3]])
eigenvalues, _ = np.linalg.[1](A)
print(eigenvalues)

Drag options to blanks, or click blank then click option'

Aeig

Bdet

Csolve

Dinv

Attempts:

3 left

💡 Hint

Common Mistakes

Using np.linalg.inv() which computes inverse, not eigenvalues.

Using np.linalg.det() which computes determinant, not eigenvalues.

The function np.linalg.eig() returns eigenvalues and eigenvectors of a square matrix.

2fill in blank

medium

Complete the code to extract only eigenvalues from the result.

NumPy

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

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Avectors

Bvals

Ceigenvectors

Dvecs

Attempts:

3 left

💡 Hint

Common Mistakes

Naming the second variable as eigenvalues again.

Using unrelated variable names like 'vals' which is unclear.

The second returned value from np.linalg.eig() is the eigenvectors, commonly named eigenvectors.

3fill in blank

hard

Fix the error in the code to correctly compute eigenvalues.

NumPy

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

Drag options to blanks, or click blank then click option'

Aeigvals

Beig

Ceigen

Deigenvalues

Attempts:

3 left

💡 Hint

Common Mistakes

Using eigvals which returns only eigenvalues and causes an unpacking error.

Using non-existent functions like eigen or eigenvalues.

The function np.linalg.eig() returns both eigenvalues and eigenvectors of a square matrix. We ignore the eigenvectors using _.

4fill in blank

hard

Fill both blanks to create a dictionary of eigenvalue to eigenvector mapping.

NumPy

import numpy as np
D = np.array([[5, 4], [1, 2]])
eigenvalues, eigenvectors = np.linalg.eig(D)
result = {eigenvalues[[1]]: eigenvectors[:, [2]] for i in range(len(eigenvalues))}
print(result)

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Ai

B0

C1

Dlen(eigenvalues)

Attempts:

3 left

💡 Hint

Common Mistakes

Using fixed indices like 0 or 1 instead of the loop variable.

Using length of eigenvalues as index which causes error.

Use the loop variable i to index eigenvalues and eigenvectors to build the dictionary.

5fill in blank

hard

Fill all three blanks to filter eigenvalues greater than 3 and create a dictionary with their eigenvectors.

NumPy

import numpy as np
E = np.array([[6, 2], [2, 3]])
eigenvalues, eigenvectors = np.linalg.eig(E)
filtered = {eigenvalues[[1]]: eigenvectors[:, [2]] for [3] in range(len(eigenvalues)) if eigenvalues[[1]] > 3}
print(filtered)

Drag options to blanks, or click blank then click option'

Ai

Bj

Ck

Dm

Attempts:

3 left

💡 Hint

Common Mistakes

Using different variables for indexing causing errors.

Using fixed indices instead of loop variable.

Use the same loop variable i to index eigenvalues and eigenvectors and filter eigenvalues greater than 3.

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