Identity matrix

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 A

Ridge 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))           # 1.0

Because of floating-point errors the results may not be exactly equal; np.allclose is used for comparison.

Exercise 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.

Show example 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)))

Test yourself

How many 1s does <code>np.eye(4)</code> contain?
- 4
- 16
- 1
- 8
Show answer

4 — Only the diagonal is 1; a 4×4 matrix has 4 diagonal elements.
Create a 3×3 identity matrix.
```
I = np.____(3)
```
Show answer
```
I = np.eye(3)
```
Match the concepts.

np.eye

Identity matrix
np.linalg.inv

Inverse matrix
np.linalg.det

Determinant
np.allclose

Approximate equality

Next lesson: Summary - Linear algebra

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