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Matrices
62 topics across 7 chapters
Chapter 1
Matrix basics & notation
1
Entries, indexing, dimensions, equality (Aij, m×n)
2
Common matrix types (square, diagonal, triangular, symmetric, orthogonal, etc.)
3
Vectors as special matrices (column/row vectors) & dot products
4
Block matrices and partitioning (how to multiply/add by blocks)
5
Transpose and (if needed) conjugate transpose
Chapter 2
Matrix arithmetic & algebra
6
Addition, subtraction, scalar multiplication (properties + practice)
7
Matrix multiplication (definition, associativity, non-commutativity, dimensions)
8
Identity matrices, invertibility, and inverse matrices
1 subtopics
9
Invertible matrix theorem (key equivalent conditions: det, rank, solutions, etc.)
10
Powers of matrices & polynomials in matrices (e.g., A^k, p(A))
11
Elementary row operations and elementary matrices
Chapter 3
Determinants, rank & trace
12
Determinants: definition, properties, and interpretation
2 subtopics
13
Determinant by expansion (small sizes) and cofactor practice
14
Geometric meaning: area/volume scaling and orientation
15
Cofactors, adjugate, and inverse via adj(A)/det(A) (when appropriate)
16
Trace and characteristic polynomial (basic identities and meaning)
17
Rank, null space, and the fundamental subspaces
2 subtopics
18
Column space, row space, null space (how to find bases conceptually)
19
Rank–nullity theorem (what it says and how to use it)
20
Matrix norms (overview): why we measure matrix size
Chapter 4
Solving linear systems with matrices
21
Augmented matrices and row operations for linear systems
22
Triangular systems and the LU idea (solve faster after factorization)
2 subtopics
23
Forward and backward substitution (solve triangular systems)
24
Pivoting idea (why zero/small pivots matter; conceptual overview)
25
Existence/uniqueness of solutions; parameterizing solution sets
26
Overdetermined systems and least squares (normal equations + geometry)
Chapter 5
Eigenvalues, eigenvectors & diagonalization
27
Eigenvalues and eigenvectors: definitions and how to compute them
2 subtopics
28
Characteristic equation practice (especially 2×2 and 3×3 cases)
29
Eigenspaces; algebraic vs geometric multiplicity (what can go wrong)
30
Diagonalization (when possible) and computing A^k efficiently
31
Complex eigenvalues of real matrices (rotation/oscillation intuition)
32
Symmetric matrices and orthogonal diagonalization
2 subtopics
33
Spectral theorem (what it guarantees and how to use it)
34
Rayleigh quotient / quadratic form intuition (link to max/min eigenvalues)
35
Jordan form (high-level idea; why non-diagonalizable matrices exist)
Chapter 6
Matrix decompositions & applications
36
LU decomposition
1 subtopics
37
Permutation matrices and PA = LU (handling row swaps)
38
QR decomposition
2 subtopics
39
Gram–Schmidt orthonormalization (concept and computations)
40
Householder reflections (why they help; conceptual overview)
41
Singular value decomposition (SVD)
2 subtopics
42
SVD meaning and geometry (axes, stretching, best low-rank approximation)
43
Moore–Penrose pseudoinverse and solving least squares via SVD
44
Applications of matrices
4 subtopics
45
Matrices as linear transformations; composition; change of basis
46
Markov chains and stochastic matrices (steady state via eigenvectors)
47
PCA intuition from SVD (data compression and directions of variance)
48
Graph adjacency matrices (paths, walks, and powers of A)
Chapter 7
Computation & practice
49
Conditioning and numerical stability (why small errors can blow up)
50
Complexity and sparse matrices (why structure matters computationally)
51
Practice set: multiply matrices, compute det/rank, solve systems, find eigenpairs
52
Software practice: matrices in Python/NumPy (create, multiply, solve, eig/SVD)
53
Common pitfalls checklist (dimension errors, order of multiplication, etc.)
54
Proof toolkit for matrices (induction, invariants under row ops, counterexamples)

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