NumPy Linear Algebra — dot, matmul, linalg explained

import numpy as np

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

# For 2D arrays, all three are equivalent
print(A @ B)           # preferred — clean syntax
print(np.matmul(A, B)) # same
print(np.dot(A, B))    # same for 2D, different for 3D+

# For 1D vectors, @ gives scalar (dot product)
u = np.array([1, 2, 3])
v = np.array([4, 5, 6])
print(u @ v)  # 32 — dot product

import numpy as np

# 2x + y = 5
# x + 3y = 10
A = np.array([[2, 1], [1, 3]])
b = np.array([5, 10])

x = np.linalg.solve(A, b)
print(x)  # [1. 3.] — x=1, y=3

# Verify
print(np.allclose(A @ x, b))  # True

# Never do this for performance
bad = np.linalg.inv(A) @ b  # numerically worse, slower

import numpy as np

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

# Eigenvalues and eigenvectors
values, vectors = np.linalg.eig(A)
print('eigenvalues:', values)   # [5. 2.]
print('eigenvectors:\n', vectors)

# Singular Value Decomposition
M = np.random.randn(5, 3)
U, s, Vt = np.linalg.svd(M, full_matrices=False)
print(U.shape, s.shape, Vt.shape)  # (5,3) (3,) (3,3)

# Frobenius norm
print(np.linalg.norm(A, 'fro'))

package io.thecodeforge.linalg;

import org.springframework.stereotype.Service;

/**
 * Orchestrates high-performance linear algebra operations.
 * In a real-world scenario, this would interface with native libraries (BLAS/LAPACK).
 */
@Service
public class LinearAlgebraService {

    public double[] solveSystem(double[][] matrixA, double[] vectorB) {
        // Production implementations would use ND4J for NumPy-like performance
        System.out.println("Initiating LU Decomposition for system size: " + matrixA.length);
        // Mock response for architectural demonstration
        return new double[]{1.0, 3.0};
    }

    public void computeSVD(double[][] matrix) {
        // SVD logic here
        System.out.println("Computing Singular Value Decomposition...");
    }
}

# Dockerfile for high-performance math workloads
FROM python:3.11-slim

LABEL org.thecodeforge.vendor="TheCodeForge"

# Install system-level optimized BLAS libraries
RUN apt-get update && apt-get install -y --no-install-recommends \
    libopenblas-dev \
    liblapack-dev \
    && rm -rf /var/lib/apt/lists/*

WORKDIR /app
COPY requirements.txt .
RUN pip install --no-cache-dir numpy scipy

COPY . .

CMD ["python", "solve_system.py"]

-- Persistence schema for high-dimensional metadata
CREATE TABLE matrix_solutions (
    solution_id UUID PRIMARY KEY DEFAULT gen_random_uuid(),
    matrix_name VARCHAR(255) NOT NULL,
    solution_vector JSONB NOT NULL, -- Storing coordinates for flexibility
    is_stable BOOLEAN DEFAULT TRUE,
    computed_at TIMESTAMP WITH TIME ZONE DEFAULT CURRENT_TIMESTAMP
);

-- Example insertion for a solved 2D system
INSERT INTO matrix_solutions (matrix_name, solution_vector)
VALUES ('linear_system_01', '[1.0, 3.0]');