Mastering NumPy in Python: The Backbone of Scientific Computing

Chapter 5: Linear Algebra and Matrix Computations in NumPy

🔹 1. Introduction

Linear algebra forms the core of many modern applications — from machine learning and computer graphics to engineering simulations and data transformations. NumPy provides an efficient and intuitive set of tools to perform linear algebra operations using the numpy.linalg module.

In this chapter, we’ll dive into:

• Dot products and matrix multiplication
• Transposes and inverses
• Determinants and rank
• Solving systems of linear equations
• Eigenvalues and eigenvectors

These techniques are essential for data scientists, analysts, engineers, and developers working on predictive models or simulations.

🔹 2. Matrix Representation in NumPy

import numpy as np

A = np.array([[1, 2], [3, 4]])

B = np.array([[5, 6], [7, 8]])

Here, both A and B are 2×2 matrices.

🔹 3. Matrix Multiplication & Dot Product

✅ Matrix Multiplication

Use @ or np.dot() for matrix multiplication:

result = A @ B

# or

result = np.dot(A, B)

Output:

[[19 22]

 [43 50]]

✅ Element-wise Multiplication (not matrix multiplication)

A * B

# [[ 5 12]

#  [21 32]]

🔹 4. Transpose of a Matrix

Flip rows to columns:

A.T

# [[1 3]

#  [2 4]]

🔹 5. Inverse of a Matrix

To find the inverse of a square matrix A, use:

inv_A = np.linalg.inv(A)

Only works if the matrix is non-singular (det ≠ 0).

🔹 6. Identity Matrix

I = np.eye(3)

Output:

lua

Copy

[[1. 0. 0.]

 [0. 1. 0.]

 [0. 0. 1.]]

Useful for checking if:
A @ np.linalg.inv(A) ≈ I

🔹 7. Determinant of a Matrix

np.linalg.det(A)

# Output: -2.0 (means A is invertible)

🔹 8. Rank of a Matrix

Use:

np.linalg.matrix_rank(A)

🔹 9. Solving Systems of Equations

2x + 3y = 8

4x + 9y = 20

✅ Represent as AX = B

A = np.array([[2, 3], [4, 9]])

B = np.array([8, 20])

X = np.linalg.solve(A, B)

print(X)  # Solution: [2. 1.]

🔹 10. Eigenvalues and Eigenvectors

Eigen decomposition helps in PCA, dynamics, and dimensionality reduction.

vals, vecs = np.linalg.eig(A)

print("Eigenvalues:", vals)

print("Eigenvectors:\n", vecs)

🔹 11. Norms and Condition Numbers

✅ Norm (magnitude of vector)

v = np.array([3, 4])

np.linalg.norm(v)  # Output: 5.0

✅ Condition Number (matrix stability)

np.linalg.cond(A)

A high condition number means the matrix is close to singular → numerical instability.

🔹 12. Pseudoinverse

Used when the matrix isn’t square or isn’t invertible.

np.linalg.pinv(A)

🔹 13. Summary Table

🔹 14. Real-World Example: PCA (Simplified)

X = np.array([[2.5, 2.4],

              [0.5, 0.7],

              [2.2, 2.9]])

# Step 1: Mean Center

X_centered = X - np.mean(X, axis=0)

# Step 2: Covariance matrix

cov = np.cov(X_centered.T)

# Step 3: Eigen decomposition

vals, vecs = np.linalg.eig(cov)

Principal components = Eigenvectors
Importance = Eigenvalues