Modern science and engineering rely on solving large-scale linear systems, eigenvalue problems, and singular value decompositions efficiently. This book provides a practical, code-driven guide to modern numerical algorithms for tackle these high-dimensional matrix problems, with a strong focus on iterative methods, Krylov subspace projections, and randomized techniques. What You Will Learn: ¿ Linear Systems: Core iterative solvers including Conjugate Gradient (CG) and GMRES, paired with preconditioning strategies. ¿ Eigenvalue Problems: Essential algorithms spanning Power Iteration, QR, Lanczos, and Divide-and-Conquer approaches. ¿ Singular Value Decomposition: High-impact SVD techniques, including Golub-Kahan Bidiagonalization, Randomized SVD, and the Randomized Nyström Method. An Interactive, Hands-On Approach: Designed for interactive learning, this text seamlessly bridges theory, mathematical formulation, and executable Python code within a Jupyter Notebook framework. Readers can directly execute code, tweak parameters, and analyze convergence through worked-out examples and real-world computational case studies. Who This Book Is For: An ideal resource for undergraduate and graduate students, researchers, and practicing engineers in scientific computing, computational engineering, applied mathematics, and data science seeking to master modern large-scale matrix solvers.