np.eye() for identity matrices in NumPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the function np.eye(n) do in NumPy?

It creates an identity matrix of size n x n. This matrix has 1s on the diagonal and 0s elsewhere.

beginner

How can you create a 4x4 identity matrix using np.eye()?

Use np.eye(4). This returns a 4 by 4 matrix with 1s on the diagonal and 0s elsewhere.

intermediate

What is the difference between np.eye(n) and np.identity(n)?

np.eye(n) can create rectangular matrices by specifying rows and columns, while np.identity(n) only creates square identity matrices of size n x n.

intermediate

How do you create a 3x5 matrix with 1s on the diagonal using np.eye()?

Use np.eye(3, 5). This creates a matrix with 3 rows and 5 columns, with 1s on the diagonal and 0s elsewhere.

intermediate

What parameter in np.eye() controls the diagonal offset?

The k parameter controls which diagonal to place the 1s on. k=0 is the main diagonal, k>0 is above, and k<0 is below the main diagonal.

What does np.eye(3) produce?

A A 3x3 matrix of ones

B A 3x3 matrix of zeros

C A 3x3 identity matrix

D A 3x3 diagonal matrix with twos

np.eye(3) creates a 3x3 identity matrix with 1s on the diagonal and 0s elsewhere.

How do you create a 2x4 matrix with 1s on the diagonal using np.eye()?

A np.eye(2, 4)

B np.eye(4, 2)

C np.eye(2)

D np.eye(4)

np.eye(2, 4) creates a matrix with 2 rows and 4 columns with 1s on the diagonal.

What does the parameter k in np.eye(n, m, k) do?

A Sets the diagonal offset where 1s are placed

B Sets the number of rows

C Sets the number of columns

D Sets the data type

The k parameter shifts the diagonal where 1s appear: 0 is main diagonal, positive is above, negative is below.

Which function only creates square identity matrices?

A np.zeros()

B np.identity()

C np.eye()

D np.ones()

np.identity() creates only square identity matrices, while np.eye() can create rectangular ones.

What will np.eye(3, 3, k=1) produce?

A A 3x3 matrix with 1s below the main diagonal

B A 3x3 identity matrix

C A 3x3 matrix of zeros

D A 3x3 matrix with 1s on the diagonal above the main diagonal

Setting k=1 places 1s on the diagonal just above the main diagonal.

Explain how to create an identity matrix and how to adjust its size and shape using np.eye().

Think about rows, columns, and diagonal offset.

Describe the difference between np.eye() and np.identity().

Focus on shape flexibility.

Practice

1. What does the function np.eye(4) create in NumPy?

easy

A. A 4x4 identity matrix with 1s on the main diagonal

B. A 4x4 matrix filled with zeros

C. A 4x4 matrix filled with ones

D. A 4x4 diagonal matrix with 1s on the first upper diagonal

Solution

1. Step 1: Understand np.eye() default behavior

   By default, np.eye(n) creates an n x n identity matrix with 1s on the main diagonal and 0s elsewhere.

2. Step 2: Apply np.eye(4)

   Calling np.eye(4) creates a 4x4 matrix with 1s on the main diagonal and 0s elsewhere.

3. Final Answer:

   A 4x4 identity matrix with 1s on the main diagonal -> Option A

4. Quick Check:

   np.eye(4) = 4x4 identity matrix [OK]

Hint: np.eye(n) makes nxn identity matrix by default [OK]

Common Mistakes:

• Confusing np.eye() with np.ones() or np.zeros()
• Thinking np.eye() fills all elements with 1
• Assuming 1s are on an off-diagonal by default

2. Which of the following is the correct syntax to create a 3x5 identity matrix with 1s on the main diagonal using np.eye()?

easy

A. np.eye(5, 3)

B. np.eye(3, 5)

C. np.eye(3)

D. np.eye(5)

Solution

1. Step 1: Understand np.eye() parameters

   The first parameter is the number of rows, the second is columns. So np.eye(3, 5) creates a 3x5 matrix.

2. Step 2: Check the options

   np.eye(3, 5) correctly uses np.eye(3, 5) to create a 3x5 matrix with 1s on the main diagonal.

3. Final Answer:

   np.eye(3, 5) -> Option B

4. Quick Check:

   np.eye(rows, cols) = correct syntax [OK]

Hint: First arg = rows, second = columns in np.eye() [OK]

Common Mistakes:

• Swapping rows and columns parameters
• Using only one parameter for non-square matrix
• Confusing np.eye() with np.identity()

3. What is the output of the following code?

import numpy as np
print(np.eye(3, k=1))

medium

A. [[0. 0. 1.]
[0. 1. 0.]
[1. 0. 0.]]

B. [[1. 0. 0.]
[0. 1. 0.]
[0. 0. 1.]]

C. [[0. 0. 0.]
[1. 0. 0.]
[0. 1. 0.]]

D. [[0. 1. 0.]
[0. 0. 1.]
[0. 0. 0.]]

Solution

1. Step 1: Understand the k parameter in np.eye()

   The parameter k=1 shifts the diagonal of 1s one above the main diagonal.

2. Step 2: Visualize the 3x3 matrix with 1s on the first upper diagonal

   The 1s appear at positions (0,1) and (1,2), zeros elsewhere.

3. Final Answer:

   [[0. 1. 0.] [0. 0. 1.] [0. 0. 0.]] -> Option D

4. Quick Check:

   k=1 shifts diagonal up by 1 [OK]

Hint: k shifts diagonal: 0=main, 1=above, -1=below [OK]

Common Mistakes:

• Ignoring the k parameter effect
• Assuming k=1 means main diagonal
• Confusing rows and columns in output

4. The code below throws an error. What is the mistake?

import numpy as np
matrix = np.eye(4, 3, k=4)
print(matrix)

medium

A. np.eye() does not accept three arguments

B. The matrix must be square for np.eye()

C. The value of k is too large for the matrix shape

D. The print statement is incorrect

Solution

1. Step 1: Check the meaning of k relative to matrix size

   The parameter k=4 means the diagonal of 1s is shifted 4 above the main diagonal.

2. Step 2: Compare k with matrix dimensions

   For a 4x3 matrix, the maximum upper diagonal shift is 2 (columns - 1). k=4 is invalid and causes an error.

3. Final Answer:

   The value of k is too large for the matrix shape -> Option C

4. Quick Check:

   k must be within valid diagonal range [OK]

Hint: k must fit within matrix diagonal limits [OK]

Common Mistakes:

• Assuming any k value works
• Thinking np.eye() only creates square matrices
• Blaming print instead of parameter error

5. You want to create a 5x5 matrix with 1s on the diagonal just below the main diagonal using np.eye(). Which code achieves this and what is the resulting matrix?

hard

A. np.eye(5, k=-1) resulting in 1s at positions (1,0), (2,1), (3,2), (4,3)

B. np.eye(5, k=1) resulting in 1s at positions (0,1), (1,2), (2,3), (3,4)

C. np.eye(5) resulting in 1s on the main diagonal

D. np.eye(5, k=-2) resulting in 1s two diagonals below the main

Solution

1. Step 1: Understand k parameter for diagonal shift

   Negative k shifts the diagonal of 1s below the main diagonal. k=-1 means one diagonal below.

2. Step 2: Apply np.eye(5, k=-1)

   This creates a 5x5 matrix with 1s at positions (1,0), (2,1), (3,2), (4,3) and zeros elsewhere.

3. Final Answer:

   np.eye(5, k=-1) with 1s just below main diagonal -> Option A

4. Quick Check:

   k=-1 shifts diagonal down by 1 [OK]

Hint: Use negative k for diagonals below main [OK]

Common Mistakes:

• Using positive k for below main diagonal
• Confusing positions of 1s in matrix
• Assuming np.eye() only creates main diagonal