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Identity Matrices & Diagonal Arrays (eye, identity, diag)0%
Identity Matrices & Diagonal Arrays (eye, identity, diag)
Beginner12 min read•Updated: Jul 11, 2026
Identity Matrices & Diagonal Arrays (eye, identity, diag)
In linear algebra, statistics, and machine learning, square matrices with ones along the main diagonal—known as Identity Matrices—act as the multiplicative identity element (analogous to the number 1 in scalar arithmetic). NumPy provides dedicated functions to construct identity matrices and manipulate matrix diagonals.
1. Creating Identity Matrices: np.eye() vs np.identity()
Both functions generate matrices with ones along the diagonal and zeros elsewhere, but np.eye() is significantly more flexible:
import numpy as np
# np.identity(n) -> Strict square N x N matrix
I_square = np.identity(3)
print(I_square)
# [[1. 0. 0.]
# [0. 1. 0.]
# [0. 0. 1.]]
# np.eye(N, M, k) -> Supports non-square shapes and diagonal offsets (k)!
E_rect = np.eye(N=3, M=4)
print(E_rect)
# [[1. 0. 0. 0.]
# [0. 1. 0. 0.]
# [0. 0. 1. 0.]]
Click to Zoom2. Offsetting Diagonals with the 'k' Parameter
In np.eye(), the k parameter shifts the diagonal of ones:
k = 0(Default): Main diagonal.k > 0(Positive): Shifts the diagonal above the main diagonal (super-diagonal).k < 0(Negative): Shifts the diagonal below the main diagonal (sub-diagonal).
# Shift diagonal 1 position UP (k=1)
print(np.eye(4, k=1, dtype=int))
# [[0 1 0 0]
# [0 0 1 0]
# [0 0 0 1]
# [0 0 0 0]]
# Shift diagonal 1 position DOWN (k=-1)
print(np.eye(4, k=-1, dtype=int))
# [[0 0 0 0]
# [1 0 0 0]
# [0 1 0 0]
# [0 0 1 0]]
3. The Dual-Role of np.diag()
The np.diag() function performs two completely different operations depending on what you pass into it:
Behavior A: Extracting the Diagonal from a 2D Matrix
When passed a 2D matrix, it extracts the diagonal elements into a 1D vector:
matrix = np.array([
[10, 20, 30],
[40, 50, 60],
[70, 80, 90]
])
diag_elements = np.diag(matrix)
print(diag_elements) # [10 50 90]
Behavior B: Constructing a 2D Matrix from a 1D Vector
When passed a 1D vector, it constructs a 2D square matrix with those values along the diagonal and zeros everywhere else:
# Create diagonal covariance / scaling matrix
scaling_matrix = np.diag([5, 10, 15])
print(scaling_matrix)
# [[ 5 0 0]
# [ 0 10 0]
# [ 0 0 15]]
Interactive Quiz
Multiple Choice Questions
Question 1
What is the multiplicative property of an identity matrix 'I' when multiplied with a compatible matrix 'A': 'A @ I'?
Question 2
What is the key functional difference between 'np.identity()' and 'np.eye()'?
Question 3
What will 'np.diag([4, 7, 9])' return?
Question 4
What will 'np.eye(3, k=-1, dtype=int)' place along the main diagonal (index [0,0], [1,1], [2,2])?
Question 5
If matrix 'M' is a 4x4 matrix, what is the shape of the output returned by 'np.diag(M)'?
Next Lesson
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Related Lessons
| Previous Lesson | Next Lesson |
|---|---|
| Inspecting Array Attributes (shape, ndim, size, itemsize) | 1D Array Indexing, Slicing & Step Strides |
Practice Quiz
Test your understanding of this lesson with 5 questions. Each question has one correct answer.