Identity matrix
- Machine Learning course
- Beginner
- Lesson 35/112
- 5 min
An identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere.
Identity matrix
The identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. It is like 1 among numbers: A @ I = A.
I = np.eye(3)
print(I)
# [[1. 0. 0.]
# [0. 1. 0.]
# [0. 0. 1.]]
A = np.array([[2, 1], [0, 3]])
print(A @ np.eye(2)) # the same as ARidge regression writes XᵀX + αI; adding a small value to the diagonal makes the solution stable.
Inverse matrix
The inverse A⁻¹ of A satisfies A @ A⁻¹ = I. Only square, non-singular (non-zero determinant) matrices have an inverse.
A = np.array([[2.0, 1.0], [1.0, 1.0]])
Ai = np.linalg.inv(A)
print(Ai) # [[ 1. -1.] [-1. 2.]]
print(np.allclose(A @ Ai, np.eye(2))) # True
print(np.linalg.det(A)) # 1.0Because of floating-point errors the results may not be exactly equal; np.allclose is used for comparison.
Practice Verify the inverse
Take the inverse of A with np.linalg.inv and print it. Then print whether A @ Ai is close to the identity matrix using np.allclose and np.eye.
Starter code
import numpy as np
A = np.array([[4.0, 7.0], [2.0, 6.0]])
# buraya yazSample solution
import numpy as np
A = np.array([[4.0, 7.0], [2.0, 6.0]])
Ai = np.linalg.inv(A)
print(Ai)
print(np.allclose(A @ Ai, np.eye(2)))Expected output
[[ 0.6 -0.7] [-0.2 0.4]] True
Questions and answers
Review the lesson questions together with their answers. To test yourself, solve the lesson interactively.
1 How many 1s does <code>np.eye(4)</code> contain?
- 4 Correct answer
- 16
- 1
- 8
Explanation: Only the diagonal is 1; a 4×4 matrix has 4 diagonal elements.
2 Create a 3×3 identity matrix.
I = np.____(3)I = np.eye(3)Explanation: np.eye(n) produces an n×n identity matrix.
3 Match the concepts.
| Term | Meaning |
|---|---|
np.eye |
Identity matrix |
np.linalg.inv |
Inverse matrix |
np.linalg.det |
Determinant |
np.allclose |
Approximate equality |