Linear Algebra — Matrices & Decomposition

Matrix Multiplication

The fundamental operation in machine learning and data science.

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

# Element-wise multiplication
A * B                    # [[5, 12], [21, 32]]

# Matrix multiplication (dot product)
A @ B                    # or np.dot(A, B)
# [[1*5 + 2*7, 1*6 + 2*8],
#  [3*5 + 4*7, 3*6 + 4*8]]

Why This Matters

• Neural networks use matrix multiplication for forward pass
• Linear regression solves: X @ w = y
• Dimensionality reduction (PCA) uses matrix decomposition

Solving Linear Systems

# Ax = b  →  x = A^(-1) @ b
A = np.array([[1, 2], [3, 4]])
b = np.array([5, 6])
x = np.linalg.solve(A, b)

Eigenvalues & Eigenvectors

Used in PCA, spectral clustering, and network analysis:

evals, evecs = np.linalg.eig(A)  # eigenvalues and eigenvectors

Common Matrix Decompositions