Advanced Matrix Mathematics Course

Lesson 1 • Introduction to Matrix Multiplication

Establishes the dot-product definition of matrix multiplication and dimension rules. Prepares students for linear transformations and system solving.

Lesson 2 • Matrix Notation and Terminology

Introduces matrix structure, indexing, and standard notation. Establishes vocabulary used throughout all subsequent chapters.

Lesson 3 • Basic Matrix Operations

Covers addition, subtraction, and scalar multiplication with formal rules. Provides computational foundation for matrix products and transformations.

Lesson 4 • Types of Matrices

Classifies matrices by structure and symmetry properties. Recognising types accelerates selection of appropriate solution methods.

Lesson 1 • Computing the Matrix Inverse

Derives inverse via adjugate formula and Gauss-Jordan elimination. Inverse computation underpins closed-form solutions in applied mathematics.

Lesson 2 • Determinant Definition and Properties

Introduces determinants via cofactor expansion and key algebraic properties. Properties reduce computation time for large matrices.

Lesson 3 • Singular vs. Invertible Matrices

Connects determinant value to matrix rank and invertibility. Distinguishes solvable from unsolvable linear systems.

Lesson 4 • Cramer's Rule and Applications

Applies determinants to solve small linear systems analytically. Connects determinant theory to practical system-solving scenarios.

Lesson 1 • LU Decomposition for System Solving

Factors matrices into lower and upper triangular form for efficient repeated solving. LU decomposition is foundational in numerical computing.

Lesson 2 • Gaussian Elimination

Applies forward elimination to reduce systems to row echelon form. Core algorithm for solving large-scale linear systems.

Lesson 3 • Solution Types and Geometric Interpretation

Classifies systems as consistent unique, consistent infinite, or inconsistent. Geometric view connects algebra to hyperplane intersections.

Lesson 4 • Reduced Row Echelon Form

Extends elimination to fully reduced form for direct solution reading. RREF reveals solution structure including free variables.

Lesson 1 • Subspaces and Spanning Sets

Identifies subspaces via closure tests and constructs spanning sets. Spanning sets describe all reachable vectors from a given collection.

Lesson 2 • Null Space and Rank-Nullity Theorem

Computes the null space of a matrix and applies the rank-nullity theorem. Connects solution structure of linear systems to subspace dimensions.

Lesson 3 • Vector Space Axioms

Defines vector spaces through closure, associativity, and distributivity axioms. Axiomatic understanding enables generalisation beyond Euclidean space.

Lesson 4 • Linear Independence and Dependence

Tests vector sets for independence using determinants and row reduction. Independence is prerequisite for constructing valid bases.

Lesson 1 • Complex Eigenvalues and Rotation

Handles complex conjugate eigenvalue pairs and their geometric meaning. Complex eigenvalues encode rotation-scaling behaviour in real matrices.

Lesson 2 • Computing Eigenvectors

Finds eigenvectors by solving the null space of (A−λI). Eigenvectors define invariant directions under linear transformation.

Lesson 3 • Diagonalisation of Matrices

Constructs diagonal form A=PDP⁻¹ when eigenvectors form a basis. Diagonalisation simplifies matrix powers and exponentials.

Lesson 4 • Characteristic Polynomial

Derives eigenvalues by solving the characteristic equation det(A−λI)=0. Polynomial roots determine the spectral structure of a matrix.

Lesson 1 • Cholesky Decomposition

Factors symmetric positive definite matrices as A=LLᵀ. Cholesky is twice as efficient as LU for symmetric systems.

Lesson 2 • QR Decomposition

Factors a matrix into orthogonal Q and upper triangular R via Gram-Schmidt. QR is central to least squares and eigenvalue algorithms.

Lesson 3 • Singular Value Decomposition Theory

Derives SVD as A=UΣVᵀ and interprets singular values geometrically. SVD is the most general and powerful matrix factorisation.

Lesson 4 • Schur Decomposition and Jordan Form

Introduces Schur triangularisation and Jordan canonical form for non-diagonalisable matrices. These forms handle defective matrices in advanced analysis.

Lesson 1 • Pseudoinverse and Generalised Solutions

Extends least squares to underdetermined and rank-deficient systems via pseudoinverse. Pseudoinverse provides minimum-norm solutions in all cases.

Lesson 2 • Inner Products and Orthogonality

Defines inner products, norms, and orthogonality in general vector spaces. Inner product structure enables projection and distance measurement.

Lesson 3 • Gram-Schmidt Process

Converts any basis into an orthonormal basis via sequential projection removal. Orthonormal bases simplify computation and improve numerical stability.

Lesson 4 • Least Squares Problem

Solves overdetermined systems by minimising residual norm. Least squares is the foundation of regression and data fitting.

Lesson 1 • Krylov Subspace Methods

Develops GMRES and Lanczos algorithms for large eigenvalue and linear problems. Krylov methods are the standard in modern scientific computing.

Lesson 2 • Matrix Exponential and Functions

Defines matrix exponential via series and diagonalisation, extending to general functions. Matrix functions solve differential equations and model dynamic systems.

Lesson 3 • Iterative Methods for Large Systems

Introduces Jacobi, Gauss-Seidel, and conjugate gradient methods for sparse systems. Iterative methods scale to problems where direct methods are infeasible.

Lesson 4 • Perturbation Theory and Sensitivity

Analyses how eigenvalues and solutions change under matrix perturbations. Sensitivity analysis guides robust algorithm design and error estimation.