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Linear Algebra Tools for Data Science Applications

About

This repository provides tools and scripts developed in Python to solve a variety of linear algebra and matrix theory problems. The goal is to deliver easy-to-use solutions for both academic and professional applications. These tools are particularly useful in data science workflows, supporting tasks such as:

• Singular Value Decomposition (SVD)
• Principal Component Analysis (PCA)
• Linear Regression
• Least Squares Optimization

Features

• Efficient Algorithms:
  • Solve linear equations quickly and accurately.
• Matrix Operations:
  • Perform addition, multiplication, inversion, and other standard matrix computations.
• Eigenvalue and Eigenvector Computation:
  • Tools to compute and analyze eigenvalues and eigenvectors.
• Complex Matrix Operations:
  • Support for advanced operations involving complex matrices.

Examples (Least Squares)

import numpy as np

def least_squares(A, b, param):
        # b = b.reshape(-1, 1)

        n = A.shape[1]

        # make [A, sqrt(param)I] and [b, 0]
        A_prime = np.concatenate((A, np.sqrt(param) * np.eye(n)), axis=0)
        b_prime = np.concatenate((b, np.zeros(n)), axis=0)

        # use the least squares solver
        return np.linalg.lstsq(A_prime, b_prime, rcond=None)[0]

def main():
        A = np.array([[1, 2], [3, 4], [5, 6]])
        b = np.array([1, 2, 3])
        param = 0.1

        actual_solution = np.linalg.inv(A.T @ A + param * np.eye(A.shape[1])) @ A.T @ b
 
        print("Least Squares Solution: ")
        print(least_squares(A, b, param))
        print("Actual Solution: ")
        print(actual_solution)

if __name__ == "__main__":
        main()

Applications

These tools can be integrated into various data science workflows, enhancing tasks such as dimensionality reduction, regression analysis, and optimization problems.