Matrix Course

Lesson 1 • Matrix Transposition

Explains the transpose operation and its algebraic properties. Transposition recurs in decompositions, least-squares methods, and covariance matrices.

Lesson 2 • Matrix Notation and Terminology

Introduces standard notation for rows, columns, and elements. Establishes the vocabulary used throughout all subsequent chapters.

Lesson 3 • Basic Arithmetic on Matrices

Teaches addition, subtraction, and scalar multiplication with conformability rules. These operations are prerequisites for matrix products and transformations.

Lesson 1 • Properties of Matrix Products

Examines associativity, distributivity, and identity element behaviour. Understanding these properties enables algebraic manipulation of matrix expressions.

Lesson 2 • Special Matrix Products

Covers outer products, Hadamard products, and Kronecker products. Each product type appears in distinct application domains covered later.

Lesson 3 • Rules of Matrix Multiplication

Derives the row-by-column rule and dimension requirements for valid products. Correct application prevents the most common computational errors.

Lesson 1 • Computing Determinants

Presents cofactor expansion and row-reduction methods for determinants. Accurate computation is essential for invertibility tests and Cramer's rule.

Lesson 2 • Computing the Matrix Inverse

Teaches adjugate-based and row-reduction methods for finding inverses. Inverse computation enables direct solution of linear equations and transformations.

Lesson 3 • Singular vs. Invertible Matrices

Defines singularity through determinant and rank criteria. Distinguishing invertible from singular matrices is critical for solving linear systems.

Lesson 4 • Cramer's Rule and Applications

Applies determinants to solve small linear systems via Cramer's rule. Connects determinant theory to practical equation-solving workflows.

Lesson 1 • Gauss-Jordan Elimination and RREF

Extends elimination to reduced row echelon form for direct solution reading. RREF also reveals rank and free variables in underdetermined systems.

Lesson 2 • Matrix Representation of Linear Systems

Converts equation systems into augmented matrix form. This representation is the entry point for all algorithmic solution methods.

Lesson 3 • Gaussian Elimination

Applies elementary row operations to reach row echelon form. Gaussian elimination is the foundational algorithm for solving linear systems.

Lesson 4 • LU Decomposition for Linear Systems

Factors a matrix into lower and upper triangular components for efficient solving. LU decomposition reduces repeated computation when solving multiple right-hand sides.

Lesson 1 • Characteristic Equation and Eigenvalues

Derives eigenvalues from the characteristic polynomial of a square matrix. Eigenvalues quantify scaling behaviour along invariant directions.

Lesson 2 • Symmetric Matrices and Spectral Theorem

Proves that symmetric matrices have real eigenvalues and orthogonal eigenvectors. The spectral theorem underpins principal component analysis and quadratic forms.

Lesson 3 • Computing Eigenvectors

Finds eigenvectors by solving the null space of shifted matrices. Eigenvectors define the invariant axes of a linear transformation.

Lesson 4 • Diagonalisation of Matrices

Constructs diagonal form using eigenvector matrices when conditions are met. Diagonalisation simplifies matrix powers and exponential computations.

Lesson 1 • Linear Transformations and Their Matrices

Defines linear maps and derives their standard matrix representations. Every matrix encodes a linear transformation between vector spaces.

Lesson 2 • Vector Space Axioms and Examples

Defines vector spaces through closure, identity, and inverse axioms. Recognising valid vector spaces is prerequisite to subspace and basis analysis.

Lesson 3 • Coordinate Systems and Change of Basis

Expresses vectors in different bases using transition matrices. Change of basis is essential for diagonalisation and efficient computation.

Lesson 4 • Subspaces, Span, and Basis

Identifies subspaces and constructs spanning sets and minimal bases. Basis selection determines coordinate representations used in transformations.

Lesson 1 • Choosing and Applying Decompositions

Provides a decision framework for selecting decompositions based on matrix properties and goals. Practical selection criteria prevent inefficient or unstable computations.

Lesson 2 • QR Decomposition

Factors a matrix into orthogonal and upper triangular components via Gram-Schmidt. QR decomposition is the backbone of least-squares solvers and eigenvalue algorithms.

Lesson 3 • Schur and Jordan Decompositions

Presents Schur triangularisation and Jordan normal form for non-diagonalisable matrices. These forms handle defective matrices that resist standard diagonalisation.

Lesson 4 • Cholesky Decomposition

Factors symmetric positive definite matrices into triangular form. Cholesky is twice as efficient as LU for positive definite systems in optimisation.

Lesson 1 • Least-Squares and Regression

Solves overdetermined systems via normal equations and pseudoinverse. Least-squares is the mathematical core of linear regression and data fitting.

Lesson 2 • Quadratic Forms and Optimisation

Analyses quadratic objective functions using matrix representations and eigenvalues. Quadratic form analysis determines convexity and optimality conditions.

Lesson 3 • Matrix Methods in Network Analysis

Encodes graphs as adjacency and Laplacian matrices and extracts structural properties. Spectral graph theory connects eigenvalues to connectivity and clustering.

Lesson 4 • Iterative Solvers and Convergence

Introduces Jacobi, Gauss-Seidel, and conjugate gradient methods for large sparse systems. Iterative methods scale where direct decompositions become computationally prohibitive.