Topic Index

Concepts mapped to the chapters where they are introduced and developed. Chapter numbers in bold mark where a topic is introduced or treated in greatest depth; the others are where it returns. Notation entries (symbols, Greek letters) are collected at the end. For a definition rather than a location, see the Glossary.

A

Affine transformations — Ch. 12
Algebraic multiplicity — Ch. 24, 25, 36
Angle between vectors — Ch. 18, 19, 34
Associativity (of matrix multiplication) — Ch. 8
Augmented matrix — Ch. 3, 4, 9

B

Back-substitution — Ch. 4, 10, 20
Basis — Ch. 6, 15, 16; orthonormal basis Ch. 20, 27
Bilinearity — Ch. 18, 34
Bloch sphere / qubit — see Qubit

C

Cauchy–Schwarz inequality — Ch. 18, 34
Cayley–Hamilton (characteristic polynomial) — Ch. 24
Change of basis — Ch. 16, 25, 27
Characteristic polynomial / equation — Ch. 23, 24, 26
Cholesky factorization — Ch. 28
Cofactor (Laplace) expansion — Ch. 11
Column picture (of a linear system) — Ch. 3
Column space $C(A)$ — Ch. 7, 13, 14, 17, 19, 30
Column-stochastic matrix — Ch. 25, 29
Complex eigenvalues — Ch. 26, 37
Composition of transformations — Ch. 8, 12, 33
Condition number — Ch. 9, 30, 38
Conjugate (Hermitian) transpose — Ch. 21, 27, 34
Consistent / inconsistent systems — Ch. 3, 4, 13
Cosine similarity — Ch. 18, 33
Covariance matrix — Ch. 28, 32
Cramer's rule — Ch. 11

D

Damping factor (PageRank) — Ch. 29
Defective matrices — Ch. 24, 25, 36, 37
Determinant — Ch. 1, 11; and eigenvalues Ch. 24; and SVD Ch. 30
Diagonalization ($A = PDP^{-1}$) — Ch. 16, 25, 27
Dimension — Ch. 15, 14
Distance-preserving maps — see Isometry, Orthogonal matrices
Dot product — Ch. 18, 19, 22
Dynamical systems — Ch. 25, 29, 37

E

Eckart–Young theorem — Ch. 31
Eigenbasis — Ch. 23, 25, 27
Eigenspace — Ch. 23, 24
Eigenvalues — Ch. 23, 24, 25, 26, 27, 28, 29, 37
Eigenvectors — Ch. 23, 24, 25, 27, 29
Elementary row operations — Ch. 4, 11
Embeddings (word / latent vectors) — Ch. 18, 33
Energy (quadratic forms) — Ch. 28
Explained variance — Ch. 32, 31

F

Fourier series — Ch. 22, 34
Four fundamental subspaces — Ch. 13, 14, 19, 30
Free variables — Ch. 4, 13
Frobenius norm — Ch. 30, 31
Function spaces — Ch. 5, 22, 34

G

Gaussian elimination — Ch. 4, 9, 10, 11
Gauss–Jordan elimination — Ch. 9, 4
Geometric multiplicity — Ch. 24, 25, 36
Gibbs phenomenon — Ch. 22
Golden ratio (in eigenvalues) — Ch. 23, 30
Gram–Schmidt process — Ch. 20, 34
Graph Laplacian / spectral graph theory — Ch. 40
Graphics pipeline — see Computer graphics under Homogeneous coordinates

H

Hermitian matrices — Ch. 21, 27, 34
Hessian matrix — Ch. 28
Hilbert space — Ch. 34, 40
Homogeneous coordinates (computer graphics) — Ch. 12, 39
Homogeneous systems — Ch. 3, 13

I

Idempotent (projection) matrices — Ch. 19
Identity matrix — Ch. 7, 8, 9
Image (of a linear map) — see Column space; abstract Ch. 35
Image compression (SVD) — Ch. 30, 31, 39
Inconsistent systems — see Consistent / inconsistent systems
Inner product spaces — Ch. 18, 34
Inverse matrix — Ch. 9, 11
Invertible Matrix Theorem — Ch. 9, 11, 14
Iterative methods (Jacobi, Gauss–Seidel, conjugate gradient) — Ch. 38
Isometry — Ch. 21

J

Jacobian (change-of-variables factor) — Ch. 11
Jordan normal form / Jordan blocks — Ch. 36, 37

K

Kernel — see Null space; abstract maps Ch. 35
Krylov subspace — Ch. 38

L

Latent semantic analysis — Ch. 30, 33
Law of cosines — Ch. 18
Least squares — Ch. 17, 19, 20, 30
Left null space $N(A^{\mathsf{T}})$ — Ch. 14, 30
Leslie / population models — Ch. 23, 25
Linear combination — Ch. 2, 3, 6
Linear dependence / independence — Ch. 6, 15
Linear regression — Ch. 17, 19
Linear transformations — Ch. 1, 7, 8, 35
LU and PLU decomposition — Ch. 10, 11, 38

M

Machine learning (neural nets, recommenders) — Ch. 33
Magnitude (of a vector) — see Norm
Mahalanobis distance — Ch. 28
Markov chains — Ch. 25, 29
Matrices as functions — Ch. 1, 7
Matrix addition / scalar multiplication — Ch. 8
Matrix exponential $e^{At}$ — Ch. 37
Matrix multiplication (as composition) — Ch. 8, 12
Minors and cofactors — Ch. 11
Multilinear algebra — see Tensors

N

Neural networks — Ch. 33
Non-commutativity (of matrix products) — Ch. 8
Norm — Ch. 2, 18; matrix norms Ch. 30
Normal equations — Ch. 17, 19
Normal modes (vibration) — Ch. 23, 27
Null space $N(A)$ — Ch. 6, 13, 14, 23
Nullity / rank–nullity — see Rank–nullity theorem
Numerical linear algebra — Ch. 30, 38

O

Operator (spectral) norm — Ch. 30, 31
Orientation (sign of the determinant) — Ch. 11, 21
Orthogonal complement — Ch. 14, 19
Orthogonal matrices — Ch. 21, 27, 30
Orthogonal projection — Ch. 17, 19, 22
Orthonormal basis — Ch. 20, 27, 34

P

PageRank — Ch. 3, 23, 29, 39
Parallelogram law / rule — Ch. 2, 18
Parametric solutions — Ch. 4, 13
Parseval's identity — Ch. 22, 34
Permutation matrices — Ch. 10
Perron–Frobenius theorem — Ch. 29
Phase portraits — Ch. 37
Pivots and pivot columns — Ch. 4, 13, 14
Polar decomposition — Ch. 30
Positive definite / semidefinite matrices — Ch. 28, 32
Power iteration — Ch. 23, 29
Principal component analysis (PCA) — Ch. 27, 32
Projection matrices — Ch. 19
Pseudoinverse — Ch. 30

Q

QR decomposition — Ch. 20, 38
Quadratic forms — Ch. 28
Qubit (quantum bit) — Ch. 1, 5, 21, 27, 34
Quantum gates (unitary matrices) — Ch. 21, 27

R

Random surfer model — Ch. 29
Rank — Ch. 3, 4, 14, 30
Rank-$k$ (low-rank) approximation — Ch. 31, 32, 33
Rank–nullity theorem — Ch. 14, 35
Real canonical form (complex eigenvalues) — Ch. 26
Recommendation systems / matrix factorization — Ch. 33, 39
Reduced row echelon form (RREF) — Ch. 4, 14
Reflections — Ch. 7, 21
Rotations — Ch. 1, 7, 21, 26
Row picture (of a linear system) — Ch. 3
Row space $C(A^{\mathsf{T}})$ — Ch. 14

S

Scalar multiplication — Ch. 2, 5, 8
Similarity (similar matrices) — Ch. 16, 25, 36
Singular matrices — Ch. 3, 9, 11
Singular values — Ch. 30, 31, 38
Singular value decomposition (SVD) — Ch. 30, 31, 32, 33
Span — Ch. 6, 13, 15
Spectral theorem — Ch. 27, 28, 32
Stability (of dynamical systems) — Ch. 37, 38
Standard basis vectors — Ch. 2, 7
Stationary distribution / steady state — Ch. 25, 29
Stochastic matrices — see Column-stochastic matrix
Subspaces — Ch. 6, 13
Superposition — Ch. 1, 7
Sylvester's criterion — Ch. 28
Symmetric matrices — Ch. 8, 27, 28
Systems of linear equations — Ch. 3, 4

T

Tensors / multilinear algebra — Ch. 40
Trace — Ch. 16, 23
Transition matrix — see Change of basis (Ch. 16) and Markov chains (Ch. 29)
Transpose — Ch. 7, 8
Triangle inequality — Ch. 18, 34
Triangular matrices and systems — Ch. 4, 10, 11

U

Unit vector / normalization — Ch. 18
Unitary matrices — Ch. 21, 27, 34

V

Variance maximization — Ch. 32
Vector addition — Ch. 2, 5
Vector spaces (axioms, examples) — Ch. 5, 6, 34, 35
Vectors — Ch. 2
Vibration / mass–spring systems — Ch. 23, 27, 37
Visualizer (2D transformation tool) — Ch. 1, 7, 11, 16, 21, 23, 30
Volume scaling (determinant) — Ch. 11

W

Whitening — Ch. 7, 32
Word embeddings — see Embeddings

Z

Zero vector — Ch. 2, 5, 6

Notation

$A$, $B$, $P$, $Q$ (italic capitals) — matrices — Ch. 7
$\mathbf{v}$, $\mathbf{x}$, $\mathbf{b}$ (bold lowercase) — vectors (columns by default) — Ch. 2
$\mathbf{e}_i$ — standard basis vectors — Ch. 7
$A^{\mathsf{T}}$ — transpose — Ch. 7
$A^{-1}$ — inverse — Ch. 9
$A^{*}$ — conjugate (Hermitian) transpose — Ch. 27
$\det(A)$, $\operatorname{tr}(A)$, $\operatorname{rank}(A)$ — determinant, trace, rank — Ch. 11, 23, 14
$C(A)$, $N(A)$, $C(A^{\mathsf{T}})$, $N(A^{\mathsf{T}})$ — the four fundamental subspaces — Ch. 13–14
$\operatorname{span}\{\cdots\}$ — span — Ch. 6
$\mathbf{u}\cdot\mathbf{v}$, $\langle\mathbf{u},\mathbf{v}\rangle$ — dot / inner product — Ch. 18, 34
$\lVert\mathbf{v}\rVert$ — norm — Ch. 18
$\lambda$, $\mathbf{v}$ in $A\mathbf{v} = \lambda\mathbf{v}$ — eigenvalue, eigenvector — Ch. 23
$A = PDP^{-1}$ — eigendecomposition — Ch. 25
$A = U\Sigma V^{\mathsf{T}}$ — singular value decomposition — Ch. 30
$\sigma_i$ — singular values — Ch. 30
$\kappa(A)$ — condition number — Ch. 38
$\dim(V)$ — dimension — Ch. 15
$\mathbb{R}^n$, $\mathbb{C}^n$ — real / complex $n$-space — Ch. 2, 26