Applied Linear Algebra for Research

A skill for applying linear algebra to research computing, data analysis, and scientific modeling. Covers matrix decompositions, eigenvalue problems, least squares, dimensionality reduction, and practical implementation in NumPy/SciPy.

Essential Operations

Matrix Multiplication and Solving Systems

import numpy as np
from scipy import linalg


def solve_linear_system(A: np.ndarray, b: np.ndarray) -> dict:
    """
    Solve Ax = b and analyze the system.

    Args:
        A: Coefficient matrix (n x n)
        b: Right-hand side vector (n,)
    """
    n = A.shape[0]

    # Check condition number (sensitivity to perturbations)
    cond = np.linalg.cond(A)

    result = {
        "shape": A.shape,
        "rank": np.linalg.matrix_rank(A),
        "condition_number": cond,
        "well_conditioned": cond < 1e10,
    }

    if result["rank"] == n:
        x = np.linalg.solve(A, b)
        result["solution"] = x
        result["residual_norm"] = np.linalg.norm(A @ x - b)
    else:
        # Underdetermined or singular -- use least-squares
        x, residuals, rank, sv = np.linalg.lstsq(A, b, rcond=None)
        result["least_squares_solution"] = x
        result["note"] = "System is rank-deficient; least-squares solution returned"

    return result

Matrix Decompositions

LU Decomposition (Solving Multiple Systems)

def lu_factorization(A: np.ndarray) -> dict:
    """
    LU decomposition for efficiently solving Ax=b for multiple b.
    """
    P, L, U = linalg.lu(A)

    return {
        "P": P,  # Permutation matrix
        "L": L,  # Lower triangular
        "U": U,  # Upper triangular
        "usage": (
            "Once computed, solve for any new right-hand side b "
            "in O(n^2) instead of O(n^3). Use scipy.linalg.lu_solve()."
        )
    }

Singular Value Decomposition (SVD)

def svd_analysis(A: np.ndarray) -> dict:
    """
    SVD of matrix A = U S V^T and its applications.

    Args:
        A: Input matrix (m x n)
    """
    U, s, Vt = np.linalg.svd(A, full_matrices=False)

    return {
        "U_shape": U.shape,       # Left singular vectors (m x k)
        "singular_values": s,      # Sorted descending
        "Vt_shape": Vt.shape,     # Right singular vectors (k x n)
        "rank": np.sum(s > 1e-10),
        "condition_number": s[0] / s[-1] if s[-1] > 0 else float("inf"),
        "energy_ratio": np.cumsum(s ** 2) / np.sum(s ** 2),
        "applications": [
            "Low-rank approximation (truncated SVD)",
            "Principal Component Analysis (PCA)",
            "Pseudoinverse computation",
            "Latent Semantic Analysis (LSA) in text mining",
            "Image compression",
            "Noise reduction"
        ]
    }