Notation

This page is a reference, not a section: there is nothing to work through here, and it carries no exercises and no quick checks. Use it when you meet a symbol and want to know what it means and where the book fixed it. The symbols are grouped by the kind of thing they name, not by chapter, because a reader arrives with a symbol and not with a chapter number. Every entry links to the place where the book defines it; follow the link for the hypotheses, which a one-line gloss cannot carry.

Read the conventions first. Most of what looks like a clash between two symbols is resolved by one of them.

The standing conventions

The field. $F$ is an arbitrary field throughout, and $\mathbb{F}_p$ is the field with $p$ elements (Definition 0.7.1). Chapters 11 to 13 and much of Part V need $\mathbb{R}$ or $\mathbb{C}$ , and say so where they do. Vector spaces are $V, U, W$ , plain italic, and are finite-dimensional unless the statement says otherwise.

Bold is for matrices and coordinate vectors, never for maps, spaces, or abstract vectors. A matrix is a bold upright capital, $\mathbf{A}$ ; a coordinate vector, an element of $F^n$ , is a bold lowercase letter, $\mathbf{v}$ ; an element of an abstract vector space is plain italic, $v$ , so the letter shows whether a computation is in coordinates or in the structure. The test is exact: an object is bold only when it is an element of a fixed, finite-dimensional $F^n$ or $F^{m \times n}$ viewed in coordinates. So the zero matrix is bold, $\mathbf{O}$ , as every matrix is; the zero vector is bold in $F^n$ and plain, $0$ , in an abstract space; and a sequence in $F^{\mathbb{N}}$ , a coset $v + U$ , and the quaternion units $i, j, k$ are plain. A coset’s representative keeps its own type, so in $F^n/U$ one writes $\mathbf{x} + U$ ; an operator or linear map is plain italic, $T$ ; a space is plain italic, $V$ . Entries stay plain: $a_{ij}$ , or $(\mathbf{A})_{ij}$ . A Greek letter naming a matrix is bold too, as $\boldsymbol{\Lambda}$ or $\boldsymbol{\Sigma}$ .

Maps and composition. Linear maps are $T, S \colon V \to W$ , never $f$ . Composition is written $ST$ , and it means: first $T$ , then $S$ . For functions in general the book writes $g \circ f$ .

Coordinates. $[v]_{\mathcal{B}}$ is the coordinate vector of $v$ in the ordered basis $\mathcal{B}$ (Definition 1.5.2), and $[T]_{\mathcal{B}}^{\mathcal{C}}$ is the matrix of $T$ with input basis $\mathcal{B}$ and output basis $\mathcal{C}$ (Definition 4.1.1). The subscript is the basis read into the first slot and the superscript the basis read into the second: for a map that is the input basis below and the output basis above, and for a bilinear form $[\beta]_{\mathcal{B}}^{\mathcal{B}}$ it is the basis used on the first argument below and the one used on the second above. Both slots are always written out, even when they hold the same basis. A basis is an ordered list, written in parentheses.

Transpose and adjoint. $\mathbf{A}^{\top}$ is the transpose (Definition 0.9.10) and $\mathbf{A}^{*}$ the conjugate transpose (Definition 11.1.2), which is also the symbol for the adjoint of an operator (Definition 11.6.1). The prime $T'$ is the dual map (Definition 5.4.1), and never a derivative of a map.

Inner products are linear in the first slot and conjugate-linear in the second (Definition 11.1.1). Over $\mathbb{R}$ the distinction disappears.

Ordering of eigenvalues and singular values. For Hermitian $\mathbf{A}$ , the eigenvalues are listed decreasing, $\lambda_1(\mathbf{A}) \ge \dots \ge \lambda_n(\mathbf{A})$ , with multiplicity. Singular values are listed decreasing as well, $\sigma_1(\mathbf{A}) \ge \sigma_2(\mathbf{A}) \ge \dots \ge 0$ (Definition 13.9.1). Where a proof wants the increasing order it says so.

Indices start at 1, with three announced exceptions: Chapter 12 §09, where circulant and Fourier subscripts run from $0$ to $n-1$ and are read modulo $n$ ; Chapter 22 §§06 to 10, where the coordinates of $F^{n+1}$ run from $0$ so that $\mathbb{P}^n(F)$ has dimension $n$ (Definition 22.6.2); and Chapter 23 §09, which follows Chapter 12 §09 for group elements and characters.

Defining, not asserting. $\mathrel{:=}$ means “is defined to be”. A plain $=$ asserts an equality that has been or will be proved.

Numbers, sets, and fields

Symbol	Meaning	Defined at
$\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$	natural numbers (with $0$ ), integers, rationals, reals, complex numbers	Definition 0.3.1
$F$ , $\mathbb{F}_p$	a field; the field with $p$ elements	Definition 0.7.1
$\operatorname{char} F$	characteristic of a field: the least $n > 0$ with $n \cdot 1 = 0$ , or $0$	Definition 0.7.2
$\overline{z}$ , $\lvert z \rvert$	complex conjugate and modulus	Definition 0.6.2
$\{ x \in S : P(x) \}$	set-builder notation, with a colon	Definition 0.3.2
$\emptyset$ , $S \subseteq T$	the empty set; inclusion	Definition 0.3.3, Definition 0.3.4
$S \cup T$ , $S \cap T$ , $S \setminus T$ , $S^{c}$	union, intersection, difference, complement	Definition 0.3.6, Definition 0.3.7, Definition 0.3.8
$S \times T$ , $(x_1, \dots, x_n)$	Cartesian product and $n$ -tuples	Definition 0.3.10, Definition 0.3.11
$\mathcal{P}\left( S \right)$	power set, the set of all subsets	Definition 0.3.12
$f \colon X \to Y$ , $x \mapsto f(x)$	a function and its rule	Definition 0.4.1
$f(A)$ , $f^{-1}(B)$	image of a subset, preimage of a subset	Definition 0.4.2
$g \circ f$ , $\operatorname{id}_X$ , $f^{-1}$	composition, identity function, inverse function	Definition 0.4.4, Definition 0.4.5, Definition 0.4.7
$x \sim y$ , $[x]$ , $X/{\sim}$	equivalence relation, equivalence class, quotient set	Definition 0.5.2, Definition 0.5.3, Definition 0.5.5
$S_n$ , $\operatorname{sgn}(\sigma)$ , $\operatorname{inv}(\sigma)$	symmetric group; sign of a permutation; number of inversions	Example (Groups), Definition 7.3.2, Definition 7.3.1
$A_n$	alternating group, the even permutations	Definition 7.3.3
$F[x]$ , $F[x]_{\le n}$	polynomials over $F$ ; those of degree at most $n$	Definition 0.8.2, Definition 0.8.4
$\deg p$ , $p(c)$	degree; evaluation at a scalar	Definition 0.8.3, Definition 0.8.6
$p \mid q$ , $\gcd(p,q)$ , $\operatorname{lcm}(p,q)$	divisibility, greatest common divisor, least common multiple in $F[x]$	Definition 6.1.1, Definition 6.2.2, Definition 6.2.4
$\langle p \rangle$	the ideal of $F[x]$ generated by $p$ : all multiples of $p$	Definition 6.2.1
$\operatorname{mult}_c(f)$	multiplicity of $c$ as a root of $f$	Definition 6.4.1
$p'$	formal derivative of a polynomial	Definition 6.4.3
$\ell_i$	Lagrange basis polynomial, $\ell_i(c_j) = \delta_{ij}$	Definition 6.6.1

Spaces

Symbol	Meaning	Defined at
$V, U, W$	vector spaces over $F$ ; $U$ is usually a subspace	Definition 1.1.1, Definition 1.2.1
$F^n$ , $F^{S}$	column vectors; all functions $S \to F$ . An element of $F^{S}$ , a sequence in $F^{\mathbb{N}}$ included, is plain	Definition 1.1.1
$F^{(S)}$	the free vector space on $S$ : the finitely supported functions $S \to F$	Definition 15.2.3
$M_{m \times n}(F)$ , $M_n(F)$	matrices over $F$	Definition 0.9.1
$\operatorname{span}(S)$	span; the span of the empty set is $\{0\}$	Definition 1.3.2
$\dim V$ , $\dim_F V$	dimension, over $F$ when the field matters	Definition 1.6.1
$\mathcal{B} = (v_1, \dots, v_n)$	an ordered basis; $\mathcal{E}$ is the standard basis	Definition 1.5.1, Example (Standard Bases)
$[v]_{\mathcal{B}}$	coordinate vector of $v$ in $\mathcal{B}$	Definition 1.5.2
$U + W$ , $U \oplus W$	sum of subspaces; internal direct sum	Definition 1.7.1, Definition 1.7.2
$V \times W$	product of vector spaces (the external direct sum)	Definition 4.4.1
$v + U$ , $V/U$	coset, plain even in $F^n/U$ (its representative keeps its own type: $\mathbf{x} + U$ ); quotient space	Definition 4.4.2, Definition 4.4.3
$\operatorname{codim} U$	codimension, $\dim V/U$	Definition 5.5.1
$V^{*}$ , $\varphi$	dual space; a linear functional, always a Greek letter	Definition 5.1.2, Definition 5.1.1
$\varphi_1, \dots, \varphi_n$	the dual basis of $\mathcal{B}$	Definition 5.1.3
$U^{0}$	annihilator of $U$ inside $V^{*}$	Definition 5.2.1
$V^{**}$ , $\operatorname{ev}_{v}$	double dual; evaluation at $v$	Definition 5.3.1
$V_{\mathbb{C}}$	complexification of a real space	Definition 1.9.2
$\mathcal{F}$	a family of operators in $\mathcal{L}(V)$	Definition 9.9.1

Vectors, matrices, and their operations

Symbol	Meaning	Defined at
$\mathbf{v}, \mathbf{u}, \mathbf{w}, \mathbf{x}, \mathbf{y}$	coordinate vectors, in $F^n$ : bold lowercase	Definition 1.1.2
$v, u, w, x, y$	vectors of an abstract space $V$ : plain italic	Definition 1.1.1
$\mathbf{0}$ , $0$ , $\mathbf{1}$	the zero vector of $F^n$ , bold, and of an abstract space, plain; the all-ones vector	Definition 1.1.2, Definition 9.11.1
$\mathbf{e}_1, \dots, \mathbf{e}_n$	the standard basis of $F^n$	Example (Standard Bases)
$\mathbf{E}_{ij}$	matrix unit: $1$ in entry $(i,j)$ , zeros elsewhere	Example (Standard Bases)
$\mathbf{A}, \mathbf{B}, \mathbf{C}$ , $a_{ij}$	matrices, bold upright capitals; their entries, plain	Definition 0.9.1
$\mathbf{O}_{m \times n}$ , $\mathbf{O}_n$ , $\mathbf{I}$ , $\mathbf{I}_n$	zero matrix, a bold O like every matrix ( $\mathbf{O}_n$ when square), also as a block; identity matrix	Definition 0.9.2, Definition 0.9.3
$\delta_{ij}$	Kronecker delta: $1$ when $i = j$ , else $0$	Definition 0.9.3
$\operatorname{diag}(d_1, \dots, d_n)$	the diagonal matrix with these diagonal entries	Definition 0.9.4
$\mathbf{A}^{\top}$ , $\mathbf{A}^{*}$	transpose; conjugate transpose	Definition 0.9.10, Definition 11.1.2
$\operatorname{tr} \mathbf{A}$	trace, the sum of the diagonal entries	Definition 0.9.12
$\det \mathbf{A}$	determinant	Definition 7.4.1
$\operatorname{adj} \mathbf{A}$	adjugate, the transposed matrix of cofactors	Definition 7.6.2
$M_{ij}$ , $C_{ij}$	minor and cofactor $(-1)^{i+j}M_{ij}$	Definition 7.6.1
$\mathbf{A}^{-1}$	inverse of a nonsingular matrix	Definition 0.9.13
$p(\mathbf{A})$	a polynomial evaluated at a matrix	Definition 0.9.14
$\mathbf{P}_{ij}$ , $\mathbf{D}_i(c)$ , $\mathbf{I} + c\mathbf{E}_{ij}$	the three elementary matrices: swap, scale, add	Definition 3.4.1
$\mathbf{P}_{\sigma}$	permutation matrix	Definition 3.4.2
$\mathbf{A} = \mathbf{L}\mathbf{U}$ , $\mathbf{P}\mathbf{A} = \mathbf{L}\mathbf{U}$	the LU and PLU factorizations	Definition 3.6.1, Definition 3.6.2
$\operatorname{col}(\mathbf{A})$ , $\operatorname{row}(\mathbf{A})$ , $\operatorname{nul}(\mathbf{A})$	column space, row space, null space	Definition 3.5.1, Definition 3.5.2, Definition 3.5.3
$\operatorname{rank} \mathbf{A}$ , $\operatorname{nullity} \mathbf{A}$	rank and nullity of a matrix	Definition 3.5.4, Definition 3.5.5
$\mathbf{A} \sim \mathbf{B}$	similar: $\mathbf{B} = \mathbf{P}^{-1}\mathbf{A}\mathbf{P}$	Definition 4.2.2
$\mathbf{A} \simeq \mathbf{B}$	congruent: $\mathbf{B} = \mathbf{P}^{\top}\mathbf{A}\mathbf{P}$	Definition 14.2.1
$\mathbf{A} \approx \mathbf{B}$	equivalent over $F[x]$ : $\mathbf{B} = \mathbf{U}\mathbf{A}\mathbf{V}$ with $\mathbf{U}, \mathbf{V}$ invertible over $F[x]$	Definition 10.7.2
equivalent matrices	$\mathbf{B} = \mathbf{Q}\mathbf{A}\mathbf{P}$ with $\mathbf{P}, \mathbf{Q}$ nonsingular	Definition 4.3.1
$\mathbf{A} \oplus \mathbf{B}$	block diagonal matrix with the given square diagonal blocks	Definition 8.1.2
$\mathbf{M}/\mathbf{A}$ , $\mathbf{M}/\mathbf{D}$	Schur complement of an invertible block of a block matrix	Definition 8.3.1
$\mathbf{A} \otimes \mathbf{B}$	Kronecker product	Definition 8.4.1
$\operatorname{vec} \mathbf{X}$	vectorization: the columns of $\mathbf{X}$ stacked into one column	Definition 8.4.2
$\mathbf{A} \circ \mathbf{B}$	Hadamard (entrywise) product	Definition 13.8.1
$[\mathbf{A}, \mathbf{B}] = \mathbf{A}\mathbf{B} - \mathbf{B}\mathbf{A}$	commutator	Definition 8.5.1
$\mathbf{J}$	the all-ones matrix, every entry $1$	Example (Orthogonally diagonalizing a 3 by 3 matrix)
$\mathbf{A}'(t)$ , $\mathbf{x}'(t)$	entrywise derivative of a matrix- or vector-valued function of a real variable	Definition 16.6.1
$e^{\mathbf{A}}$ , $f(\mathbf{A})$	matrix exponential; a function of a matrix through its Jordan form	Definition 10.9.2, Definition 10.9.1
$\lim_m \mathbf{A}^m$	entrywise convergence of matrices	Definition 10.10.1
$\mathbf{A} \mathbf{x} = \mathbf{b}$ , $\mathbf{A}\mathbf{x} = \mathbf{0}$	a linear system and its homogeneous version	Definition 3.1.1, Definition 3.1.2
echelon form	the staircase shapes elimination produces	Definition 3.2.3, Definition 3.2.4

Maps and their spaces

Symbol	Meaning	Defined at
$T, S \colon V \to W$	linear maps; $ST$ means first $T$ , then $S$	Definition 2.1.1
$T_{\mathbf{A}}$	the map $F^n \to F^m$ , $\mathbf{x} \mapsto \mathbf{A}\mathbf{x}$	Example (Multiplication by a Matrix)
$\operatorname{ker} T$ , $\operatorname{im} T$	kernel and image	Definition 2.2.1, Definition 2.2.2
$\operatorname{rank} T$ , $\operatorname{nullity} T$	$\dim \operatorname{im} T$ and $\dim \operatorname{ker} T$	Definition 2.3.2, Definition 2.3.1
$\mathcal{L}(V, W)$ , $\mathcal{L}(V)$	all linear maps $V \to W$ ; the operators on $V$	Definition 2.4.1
$\operatorname{End}(V)$	the same space $\mathcal{L}(V)$ , named as an algebra under composition	Definition 23.1.1
$p(T)$ , $\operatorname{id}_V$	a polynomial in an operator; the identity operator	Definition 2.4.2, Definition 2.1.1
$[T]_{\mathcal{B}}^{\mathcal{C}}$	matrix of $T$ from the input basis $\mathcal{B}$ to the output basis $\mathcal{C}$	Definition 4.1.1
$[\operatorname{id}]_{\mathcal{B}}^{\mathcal{C}}$	change-of-coordinates matrix from $\mathcal{B}$ to $\mathcal{C}$	Definition 4.2.1
$V \cong W$	isomorphic vector spaces	Definition 2.5.2, Definition 2.5.3
$T'$	dual map $W^{} \to V^{}$ ; the star is kept for adjoints	Definition 5.4.1
$\operatorname{tr} T$ , $\det T$	trace and determinant of an operator	Definition 4.5.2, Definition 7.5.1
$T\vert_U$ , $\bar T$	restriction of $T$ to an invariant subspace; the operator it induces on $V/U$	Definition 9.1.1, Definition 9.1.2
$\operatorname{GL}_n(F)$ , $\operatorname{SL}_n(F)$	general and special linear groups	Example (Groups), Theorem 7.5.9
$\mathcal{M}(V_1, \dots, V_k; W)$	multilinear maps $V_1 \times \dots \times V_k \to W$ ; the semicolon separates inputs from output	Definition 15.1.2

Inner products and norms

Symbol	Meaning	Defined at
$\langle u, v \rangle$	inner product: linear in the first slot, conjugate-linear in the second	Definition 11.1.1
$\langle \mathbf{A}, \mathbf{B} \rangle = \operatorname{tr}(\mathbf{B}^{*}\mathbf{A})$	the Frobenius inner product on matrices	Definition 11.1.1
$\left\\| v \right\\|$	norm; the induced norm $\sqrt{\langle v, v \rangle}$ unless said otherwise	Definition 11.1.3, Definition 16.1.1
$u \perp v$ , $\theta$	orthogonal vectors; the angle between two nonzero real vectors	Definition 11.1.4, Definition 11.1.5
$\mathbf{G}$	Gram matrix of a list, $(\mathbf{G})_{ij} = \langle v_j, v_i \rangle$	Definition 11.2.3
$U^{\perp}$	orthogonal complement	Definition 11.3.1
$P_U$ , $d(v, U)$	orthogonal projection onto $U$ ; distance to $U$	Definition 11.3.2, Definition 11.3.3
$T^{*}$	adjoint of an operator	Definition 11.6.1
self-adjoint, normal	$T^{} = T$ ; $T^{}T = TT^{*}$	Definition 11.6.2, Definition 11.6.3
$\operatorname{O}(n)$ , $\operatorname{SO}(n)$ , $\operatorname{U}(n)$	orthogonal, special orthogonal, and unitary groups	Definition 11.7.3
$\mathbf{H}_{\mathbf{w}}$ , $\mathbf{G}(i,j;c,s)$	Householder reflection; Givens rotation	Definition 11.8.1, Definition 11.8.2
$P_n$ , $T_n$	Legendre and Chebyshev polynomials	Definition 11.10.1, Definition 24.7.4
$\left\\| \mathbf{x} \right\\|_p$	the $p$ -norm; $q$ is the conjugate exponent, $1/p + 1/q = 1$	Definition 16.1.1, Corollary 18.11.5
$\left\\| f \right\\|_{\infty}$	sup norm on $C[0,1]$	Definition 16.1.1
$B_{\left\\| \cdot \right\\|}$ , $B_1, B_2, B_\infty$	closed unit ball of a norm; the balls of the three $p$ -norms	Definition 16.1.2
$\left\\| \mathbf{A} \right\\|_F$	Frobenius norm, $\sqrt{\operatorname{tr}(\mathbf{A}^{*}\mathbf{A})}$ ; the subscript is obligatory	Lemma 11.7.5
$\left\\| \mathbf{A} \right\\|$ , $\left\\| \mathbf{A} \right\\|_1$ , $\left\\| \mathbf{A} \right\\|_{\infty}$	operator norm induced by a vector norm; largest column sum; largest row sum	Definition 16.3.2
$\kappa(\mathbf{A}) = \left\\| \mathbf{A} \right\\|\left\\| \mathbf{A}^{-1} \right\\|$	condition number	Definition 16.8.1
$\left\\| \mathbf{y} \right\\|_{*}$	dual norm, $\max_{\left\\| \mathbf{x} \right\\| \le 1}\langle \mathbf{x}, \mathbf{y} \rangle$	Definition 18.4.4
$\left\|\!\left\|\!\left\| \mathbf{A} \right\|\!\right\|\!\right\|$ , $\left\|\!\left\|\!\left\| \mathbf{A} \right\|\!\right\|\!\right\|_{(k)}$	unitarily invariant norm; the Ky Fan $k$ -norm $\sigma_1 + \dots + \sigma_k$	Definition 21.4.1, Definition 21.4.2
$\Phi$ , $\Phi^{*}$	symmetric gauge function and its dual, with $\left\|\!\left\|\!\left\| \mathbf{A} \right\|\!\right\|\!\right\| = \Phi(\sigma(\mathbf{A}))$	Definition 21.3.1, Definition 21.3.3
$\langle \mathbf{x}, \mathbf{y} \rangle_{\mathbf{A}}$ , $\left\\| \mathbf{x} \right\\|_{\mathbf{A}}$	the inner product $\mathbf{y}^{*}\mathbf{A}\mathbf{x}$ of a positive definite $\mathbf{A}$ , and its energy norm	Theorem 13.4.2, Definition 24.7.1
$\Theta(U, W)$ , $\theta_1 \le \dots \le \theta_k$	principal angles between two subspaces, in $[0, \pi/2]$	Definition 20.8.1

Bilinear forms, tensors, and exterior algebra

Symbol	Meaning	Defined at
$\beta \colon V \times V \to F$ , $q(v) = \beta(v,v)$	bilinear form; the quadratic form it carries	Definition 14.1.1, Definition 14.3.2
$[\beta]_{\mathcal{B}}^{\mathcal{B}}$	Gram matrix of a form, $\beta(v_i, v_j)$ , indices uncrossed; $\mathcal{B}$ is written twice because each slot of the form is given a basis	Definition 14.1.2
$\operatorname{rad}(\beta)$	radical of a form; $\beta$ is nondegenerate exactly when it is $\{0\}$	Definition 14.1.3, Definition 14.1.4
discriminant	the square class of $\det[\beta]_{\mathcal{B}}^{\mathcal{B}}$ in $F^{\times}$ , the same for every basis	Definition 14.2.2
$(n_+, n_-, n_0)$	inertia of a real symmetric form; the signature is the pair $(n_+, n_-)$	Definition 14.5.1
$\operatorname{In}(\mathbf{A})$	inertia triple of a Hermitian matrix, by sign of eigenvalue	Definition 17.10.1
$\mathbf{H}_f(\mathbf{a})$	Hessian matrix of a twice-differentiable function at a point	Definition 14.6.2
isotropic, totally isotropic	$q(v) = 0$ with $v \ne 0$ ; a subspace on which $\beta$ vanishes	Definition 14.7.1, Definition 14.7.2
hyperbolic plane, $\operatorname{ind}(\beta)$	the plane of a hyperbolic pair; the Witt index	Definition 14.7.3, Definition 14.7.4
$U \perp W$	orthogonal direct sum for a form: the summands pair to zero	Definition 14.7.3
$(V,\beta) \cong (V',\beta')$	isometric bilinear spaces (this use of $\cong$ is local to forms)	Definition 14.8.1
$\operatorname{Isom}(\beta)$	isometry group of a form, $\{\mathbf{P} : \mathbf{P}^{\top}\mathbf{A}\mathbf{P} = \mathbf{A}\}$	Definition 14.10.1
$\operatorname{O}(p,q)$ , $\mathbf{I}_{p,q}$ , $\operatorname{Sp}(2m,F)$ , $\boldsymbol{\Omega}_{2m}$	the indefinite orthogonal and symplectic groups and their standard matrices	Definition 14.10.2
$V \otimes W$ , $v \otimes w$	tensor product and a simple tensor	Definition 15.2.1, Definition 15.2.2
$S \otimes T$ , $\mathcal{B} \otimes \mathcal{C}$	tensor product of maps; the product basis, first factor slow	Definition 15.3.1, Definition 15.3.2
$\Theta$	the isomorphism $V^{*} \otimes W \to \mathcal{L}(V,W)$	Theorem 15.3.9
$C$ , $C^a_b$	contraction $V^{*} \otimes V \to F$ ; contraction of one upper against one lower slot	Definition 15.4.1, Definition 15.4.3
$T^{p}_{q}(V)$ , $T^{i_1 \dots i_p}_{j_1 \dots j_q}$	mixed tensors of type $(p,q)$ and their components	Definition 15.4.2, Definition 15.5.1
$V^{\otimes k}$ , $\operatorname{T}(V)$	$k$ -th tensor power, with $V^{\otimes 0} = F$ ; the tensor algebra	Definition 15.6.1, Definition 15.6.2
$\operatorname{Sym}^{k} V$ , $\operatorname{Sym}(V)$	$k$ -th symmetric power; the symmetric algebra, written by juxtaposition	Definition 15.7.2, Definition 15.7.3
$(V^{\otimes k})^{S_k}$	the symmetric tensors, fixed by every permutation of the slots	Definition 15.7.5
$F[x_1, \dots, x_n]$	the polynomial algebra in $n$ variables	Definition 15.7.4
$\Lambda^{k} V$ , $v_1 \wedge \dots \wedge v_k$	$k$ -th exterior power and a wedge	Definition 15.8.2
$\Lambda V$ , $e_{S}$	exterior algebra; its basis element for an increasing index set	Definition 15.10.1
$\Lambda^k T$	exterior power of a linear map; on the top power it is $\det T$	Definition 15.9.1
$\varepsilon_{v}$	wedging by a fixed vector, $\omega \mapsto v \wedge \omega$	Lemma 15.9.1
$\mathcal{I}$ , $\mathcal{I}_q$	a two-sided ideal of an algebra; the one whose quotient is the Clifford algebra	Definition 15.10.3, Definition 14.12.2
$\mathbb{H}$ , $i, j, k$	the quaternions; its units, plain like the complex $i$	Definition 14.12.3

Spectra and canonical forms

Symbol	Meaning	Defined at
$p_T(x) = \det(x\mathbf{I} - T)$	characteristic polynomial, monic	Definition 7.10.1, Definition 7.10.2
$m_T(x)$	minimal polynomial: the monic generator of the annihilating ideal	Definition 9.5.1, Definition 6.7.1
$\operatorname{spec}(T)$	spectrum, the set of eigenvalues	Definition 9.2.3
$E_\lambda(T) = \operatorname{ker}(T - \lambda I)$	eigenspace	Definition 9.2.1, Definition 9.2.2
$a_T(\lambda)$ , $g_T(\lambda)$	algebraic and geometric multiplicity of $\lambda$	Definition 9.3.1, Definition 9.3.2
$G_\lambda(T)$	generalized eigenspace, $\operatorname{ker}(T - \lambda I)^{n}$	Definition 10.1.1, Definition 10.1.2
$\mathbf{J}_k(\lambda)$	$k \times k$ Jordan block: $\lambda$ on the diagonal, $1$ on the superdiagonal	Definition 10.3.1, Definition 10.3.2
$\boldsymbol{\Lambda}(\lambda)$ , $\mathbf{C}_k(\lambda)$	rotation-scaling block of a conjugate pair; the real Jordan block built from it	Definition 10.4.1
$\mathbf{C}(p)$	companion matrix of a monic $p$	Definition 10.5.3
$Z(v; T)$ , $m_{T,v}$	cyclic subspace generated by $v$ ; the $T$ -annihilator of $v$	Definition 10.5.1, Definition 10.5.2
$d_1 \mid d_2 \mid \dots \mid d_r$	invariant factors, with $d_r = m_T$ ; their prime powers are the elementary divisors	Definition 10.6.1
$\rho(\mathbf{A})$	spectral radius, the largest modulus in the spectrum	Definition 16.4.1
$\boldsymbol{\Pi}$	the limit $\lim_m \mathbf{A}^{m}$ of a convergent matrix	Theorem 10.10.4
$\mathbf{A} \succeq \mathbf{O}$ , $\mathbf{A} \succ \mathbf{O}$	positive semidefinite, positive definite; Hermitian is implied	Definition 13.1.1
$\mathbf{A} \succeq \mathbf{B}$	the Loewner order, $\mathbf{A} - \mathbf{B} \succeq \mathbf{O}$	Definition 13.5.1
$\mathbf{A}^{1/2}$	the unique positive semidefinite square root	Theorem 13.2.1
$\lvert \mathbf{A} \rvert = (\mathbf{A}^{*}\mathbf{A})^{1/2}$	absolute value of a matrix	Definition 13.2.1
$\sigma_1(\mathbf{A}) \ge \dots \ge \sigma_{\min(m,n)}(\mathbf{A})$	singular values, decreasing	Definition 13.9.1
$\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{*}$	a singular value decomposition: $\mathbf{U}$ and $\mathbf{V}$ unitary, $\boldsymbol{\Sigma}$ the same size as $\mathbf{A}$ with the singular values on its diagonal and zeros elsewhere	Theorem 13.9.1
$\mathbf{A}^{+}$	Moore–Penrose pseudoinverse	Definition 13.12.1
$\lambda_1(\mathbf{A}) \ge \dots \ge \lambda_n(\mathbf{A})$ , $\boldsymbol{\uplambda}(\mathbf{A})$	eigenvalues of a Hermitian matrix, decreasing; the vector of them	Definition 17.1.1, Definition 17.8.1
$R_{\mathbf{A}}(\mathbf{x})$ , $R_T(v)$	Rayleigh quotient $\langle \mathbf{A}\mathbf{x}, \mathbf{x} \rangle/\langle \mathbf{x}, \mathbf{x} \rangle$	Definition 17.1.1
$s_k(\mathbf{A})$	Ky Fan partial sum $\lambda_1(\mathbf{A}) + \dots + \lambda_k(\mathbf{A})$	Corollary 17.6.5
$\mathbf{d}(\mathbf{A})$	the diagonal vector $(a_{11}, \dots, a_{nn})$ of a Hermitian matrix	Definition 17.8.1
$\mathcal{W} = (W_1, \dots, W_k)$	a flag of subspaces, and a list adapted to it	Definition 17.7.1
$\mathcal{H}(\mathbf{A})$	Hermitian dilation, whose eigenvalues are $\pm\sigma_i(\mathbf{A})$ and zeros	Proposition 17.9.3
$r_i(\mathbf{A})$ , $D_i(\mathbf{A})$	Gershgorin radius and disc	Definition 20.1.1
$O_{ij}(\mathbf{A})$	Brauer oval of Cassini	Theorem 20.2.1
$W(\mathbf{A})$	numerical range, $\{\mathbf{x}^{*}\mathbf{A}\mathbf{x} : \left\\| \mathbf{x} \right\\|_2 = 1\}$	Definition 20.2.1
$\mathbf{E}$ , $\widetilde{\mathbf{A}}$	a perturbation, in $\mathbf{A} + \mathbf{E}$ ; a perturbed matrix when it is not of that shape	Definition 16.8.1
$\operatorname{sep}_F(\mathbf{A},\mathbf{B})$	separation of two matrices in the Frobenius norm	Definition 20.7.1
$\kappa_{\mathbf{A}}(\lambda)$	condition number of a simple eigenvalue	Definition 20.11.1
$\mathcal{S}_{\mathbf{A},\mathbf{B}}$	Sylvester operator $\mathbf{X} \mapsto \mathbf{A}\mathbf{X} - \mathbf{X}\mathbf{B}$	Definition 12.10.1
$\mathbf{B} - \lambda\mathbf{A}$	a matrix pencil; its generalized eigenvalues solve $\mathbf{B}\mathbf{x} = \lambda\mathbf{A}\mathbf{x}$	Definition 13.4.1

Orders, convexity, and nonnegative matrices

Symbol	Meaning	Defined at
$\mathbf{x} \ge \mathbf{0}$ , $\mathbf{A} \ge \mathbf{O}$ , $\mathbf{A} > \mathbf{O}$	the entrywise order; a positive matrix has every entry positive	Definition 19.1.1
$\mathbb{R}^m_{\ge 0}$	the nonnegative orthant	Definition 18.5.1
$\mathbf{x} \prec \mathbf{y}$ , $\mathbf{x} \prec_w \mathbf{y}$ , $\mathbf{x}^{\downarrow}$	majorization, weak majorization, decreasing rearrangement	Definition 17.8.1
$\operatorname{conv} S$ , $\operatorname{aff} S$	convex hull; affine hull, a coset of a subspace	Definition 18.1.4, Definition 18.1.5
$[x, y]$	the segment from $x$ to $y$	Definition 18.1.1
$\Delta_n$	standard simplex: $n$ nonnegative weights adding to $1$	Definition 18.1.2
$\text{int}\left( S \right)$ , $\overline{S}$	interior and closure in a finite-dimensional real space	Definition 18.3.2, Definition 18.1.6
$P_C(x)$ , $d(x, C)$	nearest point of a closed convex set, and the distance to it	Theorem 18.3.1
$p_K$ , $h_K$ , $K^{\circ}$	gauge, support function, and polar set of a convex set	Definition 18.4.1, Definition 18.4.2, Definition 18.4.3
$K^{*}$	dual cone, inside $V$ itself	Definition 18.5.3
$\operatorname{cone}(a_1, \dots, a_m)$	finitely generated cone	Definition 18.5.2
$\operatorname{epi} f$	epigraph, $\{(x, s) : s \ge f(x)\}$	Definition 18.10.1
$\operatorname{ext} C$	extreme points of a convex set	Definition 18.8.1, Definition 18.8.2
$\mathcal{D}_n$ , $\Omega_n$	real density matrices; doubly stochastic matrices	Definition 18.8.3, Definition 18.8.4
$v(\mathbf{A})$	value of the matrix game with payoff matrix $\mathbf{A}$	Definition 18.7.2, Definition 18.7.1
$\mathbf{A} \# \mathbf{B}$	geometric mean of two positive definite matrices	Definition 21.11.1
$L_f(t_1, \dots, t_n)$	Loewner matrix of divided differences	Definition 21.9.2, Definition 21.9.1
$G(\mathbf{A})$	directed graph of a square matrix: an edge $j \to i$ when $a_{ij} \ne 0$	Definition 19.2.1
irreducible, primitive	no proper invariant coordinate subspace; some power is positive	Definition 19.2.3, Definition 19.5.1
$h$ , $\omega = e^{2\pi i/h}$	period of an irreducible nonnegative matrix, and the root of unity it names	Definition 19.5.2
$\underline{r}_{\mathbf{A}}(\mathbf{x})$ , $\overline{r}_{\mathbf{A}}(\mathbf{x})$	lower and upper Collatz–Wielandt functions; the bars are not conjugation	Definition 19.4.1
Perron root and vectors	$\rho(\mathbf{A})$ , the right vector $\mathbf{v} > \mathbf{0}$ with $\mathbf{1}^{\top}\mathbf{v} = 1$ , the left vector $\mathbf{w}$ with $\mathbf{w}^{\top}\mathbf{v} = 1$	Definition 19.1.2, Definition 19.3.1
$\mathbf{Q}$ , $\mathbf{R}$ , $\mathbf{N}$	transient block, absorbing block, and fundamental matrix of an absorbing chain	Definition 19.6.1
$\mathbf{G}$ , $\mathbf{H}$ , $\alpha$	Google matrix, raw link matrix, damping factor	Definition 19.8.2, Definition 19.8.1
$\mathbf{K}_n$	the second-difference matrix, a nonsingular M-matrix	Definition 19.9.2, Definition 19.9.1
stochastic matrix, steady state	column sums $1$ and nonnegative entries; a fixed probability vector	Definition 9.11.1, Definition 9.11.2

Algebras, groups, and representations

Symbol	Meaning	Defined at
$A$ , $1_A$	an associative algebra over $F$ and its unit; plain italic, never bold	Definition 14.12.1
$\mathcal{T}_n$ , $\mathcal{N}$	the upper triangular matrices in $M_n(F)$ ; its ideal of strictly upper triangular ones	Example (Six subalgebras)
$F\langle S \rangle$	the subalgebra generated by a set: the span of the words in $S$	Definition 23.1.4
$F[T]$ , $F[\mathbf{A}]$	the algebra of polynomials in one operator or matrix, of dimension $\deg m_T$	Proposition 23.1.2
$F[G]$ , $\delta_g$	group algebra of a finite group and its basis of point masses	Definition 23.1.5
$L_a$	left multiplication by $a$ ; the regular representation is $a \mapsto L_a$	Theorem 23.1.3
$S'$ , $S''$ , $Z(A)$	commutant, double commutant, center	Definition 23.1.6, Proposition 23.1.5
$C_j$ , $L_W$ , $R_U$	the column ideals of $M_n(F)$ , and the left and right ideals of a subspace	Definition 23.2.1, Proposition 23.2.4
$A/\mathcal{I}$ , $\pi$	quotient algebra by a two-sided ideal, and the quotient homomorphism	Definition 23.2.2
$\operatorname{rad} A$	radical of an algebra: killed by every irreducible representation	Definition 23.3.5
$av$ , an $A$ -space	an action of $A$ on $V$ , that is a homomorphism $A \to \operatorname{End}(V)$	Definition 23.3.2
simple, semisimple	an $A$ -space with no proper nonzero invariant subspace; a direct sum of such	Definition 23.3.3, Definition 23.3.4
$\operatorname{End}_A(V)$ , $\operatorname{Hom}_A(V,W)$	equivariant maps, $f(av) = af(v)$ ; the subscript is never dropped	Definition 23.4.1
$A(\mathcal{F})$	the subalgebra generated by a family of operators	Lemma 23.4.2
$v\varphi$	the rank-one operator $x \mapsto \varphi(x)v$	Theorem 23.5.2
$e_i$	orthogonal idempotents of a product of algebras: italic, never bold	Definition 23.6.1
$\rho$ , $\chi_\rho(g) = \operatorname{tr}\rho(g)$	a representation of a group and its character	Definition 23.7.1, Definition 23.8.2
irreducible, equivalent	no proper nonzero subrepresentation; conjugate by one isomorphism	Definition 23.7.4, Definition 23.7.3
$\operatorname{Cl}(G)$ , $\langle f, f' \rangle$	class functions on $G$ , and their averaged inner product	Definition 23.8.3, Definition 23.8.4
$V^{G}$	the fixed space of a representation	Lemma 23.8.4
$\langle g \rangle$ , $G \times H$	the cyclic subgroup generated by $g$ ; a direct product of groups	Definition 23.9.1, Definition 23.9.2
$\widehat G$ , $\widehat f$ , $\operatorname{ev}_g$	dual group of a finite abelian group, Fourier transform, evaluation at $g$	Definition 23.9.3, Definition 23.9.5
$\mathbf{U}$	character matrix, $(\mathbf{U})_{mj} = \chi_j(g_m)/\sqrt n$ , indices from $0$	Definition 23.9.4
$\mathbf{C}$ , $\mathbf{F}$	circulant matrix; the Fourier matrix	Definition 12.9.1, Theorem 12.9.3
$\mathbf{c} \ast \mathbf{d}$	cyclic convolution, subscripts read modulo $n$	Definition 12.9.1

Geometry

Symbol	Meaning	Defined at
$\mathcal{A}, \mathcal{B}, \mathcal{C}$	affine spaces, plain calligraphic	Definition 22.1.1
$P, Q, R$ , $\overrightarrow{PQ}$ , $P + v$	points of an affine space, not bold; the vector between two points; translation of a point	Definition 22.1.1
$\mathbb{A}^n(F)$	affine $n$ -space: $F^n$ with its origin forgotten	Definition 22.1.1
$\theta_O$	the choice-of-origin bijection $\mathcal{A} \to V$ , $P \mapsto \overrightarrow{OP}$	Proposition 22.1.3
$\vec f$ , $\widehat f$	linear part of an affine map; its block matrix, homogenizing coordinate last	Theorem 22.1.5, Proposition 22.1.7
$\operatorname{Aff}(\mathcal{A})$ , $\tau_{v}$	the affine group, and translation by $v$	Theorem 22.1.6
$\vec A$ , $A \vee B$ , $A \parallel B$	direction space of a flat; the join; parallelism	Definition 22.2.1, Definition 22.2.2, Definition 22.2.3
$P_J$	the face of a polyhedron cut out by making the rows $i \in J$ tight	Proposition 22.3.1, Definition 22.3.1
$\operatorname{E}(n)$ , $t_{\mathbf{b}}$ , $\operatorname{Fix}(f)$	the Euclidean group, translation by $\mathbf{b}$ , fixed set of a motion	Definition 22.4.1
$\mathbf{R}_{\theta}$ , $\mathbf{M}_{\theta}$	plane rotation and plane reflection matrices	Theorem 11.7.6
$\widetilde{\mathbf{A}}$ , $\tilde{\mathbf{x}}$	big matrix and big vector of a quadric, with $Q(\mathbf{x}) = \tilde{\mathbf{x}}^{\top}\widetilde{\mathbf{A}}\tilde{\mathbf{x}}$	Definition 22.5.1
$\mathbb{P}(V)$ , $\mathbb{P}^n(F)$	projective space of $V$ , of dimension $\dim V - 1$ ; $\mathbb{P}(F^{n+1})$	Definition 22.6.1
$[\mathbf{x}] = [x_0 : \dots : x_n]$	homogeneous coordinates; coordinates run from $0$	Definition 22.6.2
$U_0$ , $U_j$ , $H_{\infty}$ , $\iota$	the standard affine chart, the other charts, the hyperplane at infinity, the chart bijection	Theorem 22.6.2
$S \vee T$ , $S \cap T$	join and meet of projective subspaces	Proposition 22.6.5, Definition 22.6.3
$[T]$ , $[\mathbf{A}]$	the projectivity induced by an isomorphism or an invertible matrix	Definition 22.7.1
$\operatorname{PGL}(V)$ , $F^{\times}$	projective linear group; the multiplicative group of the field	Theorem 22.7.3
$(A, B; C, D)$	cross-ratio of four points: itself a point of $\mathbb{P}^1(F)$ , not a scalar	Definition 22.7.3
$S^{0}$	projective annihilator, Chapter 5’s $U^{0}$ read in $\mathbb{P}(V^{*})$	Theorem 22.8.3, Definition 22.8.1
$o$	a chosen representative vector of a projective point	Lemma 22.9.1
$Z(q)$ , $P^{\beta}$	zero set of a quadratic form; the polar of a point with respect to a conic	Definition 22.10.1, Definition 22.10.2
$p_{ij}$ , $[L]$ , $\mathcal{K}$	Plücker coordinates of a line, the point they form, and the Klein quadric	Definition 22.10.3
$\operatorname{vol}$	volume: of a parallelepiped it is $\lvert\det\rvert$ ; extended to $k$ -dimensional volume	Definition 7.9.1, Definition 13.3.1
$(\operatorname{In}\mathbf{A}, \operatorname{In}\widetilde{\mathbf{A}})$	the pair of inertia triples of a real quadric, an invariant of its affine equivalence class	Definition 22.5.2, Definition 17.10.1

Computation

Symbol	Meaning	Defined at
$u$ , $\operatorname{fl}$ , $\mathcal{R}$ , $\delta$	unit roundoff, the rounding map, the representable numbers, one rounding	Definition 24.1.1
$\gamma_k = ku/(1-ku)$	the accumulated-rounding constant	Lemma 24.1.1
$\widehat{\mathbf{x}}$ , $\Delta\mathbf{A}$	a computed object; a data perturbation produced by an algorithm	Definition 24.1.2, Definition 24.1.3
flop, $g_n$	one arithmetic operation; the growth factor of elimination	Definition 24.2.1, Definition 24.13.1
$\mathcal{K}_k(\mathbf{A},\mathbf{b})$ , $\theta_1, \dots, \theta_k$	Krylov space $\operatorname{span}(\mathbf{b}, \mathbf{A}\mathbf{b}, \dots, \mathbf{A}^{k-1}\mathbf{b})$ ; the Ritz values from it	Definition 24.6.1, Definition 24.6.2
$\mathbf{L}$ , $\mathbf{L}_0$ , $\mu_2(\mathbf{L})$	graph Laplacian $\mathbf{D} - \mathbf{A}$ , the reduced Laplacian, the Fiedler value	Definition 24.11.1, Definition 24.11.2
$\mathbf{X}$ , $\mathbf{X}_c$ , $\mathbf{S}$	data matrix, its centered form, the sample covariance	Definition 24.10.1, Definition 24.10.2
$\mathbf{M}$ , $\mathbf{K}$ , $\omega_j$	mass matrix, stiffness matrix, natural frequencies	Definition 24.9.2
$\mathbf{W}_n$ , $\mathbf{P}_n$ , $\mathbf{D}_m$ , $\mathbf{B}_n$	unnormalized transform matrix, even-odd permutation, twiddle matrix, butterfly	Definition 24.12.1, Definition 24.12.2
$\omega$	the exponent of matrix multiplication	Definition 24.13.2

Where the book overrides itself

Twenty-four chapters do not have twenty-four alphabets. Several letters carry more than one meaning, and a few English words carry more than one definition. In every case the local meaning is declared where it is used, and in every case there is a tell — an argument, a subscript, a typeface — that decides which one is meant. The rows below list the clashes worth knowing before you meet them.

Symbol or word	The meanings	How to tell
$\mathbf{U}$	a generic upper triangular matrix (Definition 0.9.5, Definition 3.6.1); a unitary matrix, in the Schur and singular value factorizations (Corollary 12.1.3, Definition 13.9.1); the character matrix of a finite abelian group (Definition 23.9.4)	the chapter: triangular in Chapters 3 and 23, unitary from Chapter 12 on, the character matrix only in Chapter 23 §09
$\mathbf{V}$	the right factor of a singular value decomposition, $\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{*}$ (Theorem 13.9.1); the Vandermonde matrix $\mathbf{V}(x_1, \dots, x_n)$ (Theorem 7.8.1); one of the two matrices invertible over $F[x]$ in $\mathbf{B} = \mathbf{U}\mathbf{A}\mathbf{V}$ (Definition 10.7.2). Plain $V$ is a vector space	the Vandermonde matrix always carries its nodes; the typeface for the space, and Chapter 10 §07 is the one place $\mathbf{V}$ is invertible only over $F[x]$
irreducible	a polynomial with no nontrivial factorization (Definition 6.3.1); a nonnegative matrix with no invariant coordinate subspace (Definition 19.2.3); an $A$ -space or a representation with no proper nonzero invariant subspace (Definition 23.3.3, Definition 23.7.4)	what the word is applied to. The three notions are unrelated
$e_i$ versus $\mathbf{e}_i$	italic $e_i$ is an idempotent of a product of algebras (Definition 23.6.1); bold $\mathbf{e}_i$ is the $i$ -th standard basis vector (Example (Standard Bases))	the typeface, and that is the whole reason Chapter 23 chose italic
$\mathbf{G}$	Gram matrix (Definition 11.2.3); Givens rotation $\mathbf{G}(i,j;c,s)$ (Definition 11.8.2); Google matrix (Definition 19.8.2); the alternating Gram matrix of the Plücker form (Definition 22.10.3). Plain $G(\mathbf{A})$ is the directed graph of a matrix (Definition 19.2.1)	the Givens rotation always carries its four arguments; the graph is plain italic with its matrix in brackets
$\mathbf{H}$ , $\mathbf{K}$	Hessian $\mathbf{H}_f(\mathbf{a})$ (Definition 14.6.2); Householder reflection $\mathbf{H}_{\mathbf{w}}$ (Definition 11.8.1); the Hermitian and skew parts of a matrix (Lemma 20.2.3); the raw link matrix (Definition 19.8.1); the second-difference matrix $\mathbf{K}_n$ (Definition 19.9.2); the stiffness matrix (Definition 24.9.2). Calligraphic $\mathcal{K}_k(\mathbf{A},\mathbf{b})$ is a Krylov space (Definition 24.6.1) and $\mathcal{K}$ the Klein quadric (Definition 22.10.3); blackboard $\mathbb{H}$ is the quaternions (Definition 14.12.3)	the Hessian and the Householder reflection always carry their argument, the second-difference matrix always its size $\mathbf{K}_n$ , the Krylov space its subscript and arguments
$\mathbf{L}$	the lower triangular factor of an elimination (Definition 3.6.1); the graph Laplacian (Definition 24.11.1); the Lax matrix (Definition 24.8.1). Plain $L_a$ is left multiplication in an algebra (Theorem 23.1.3) and $L_f$ a Loewner matrix (Definition 21.9.2); calligraphic $\mathcal{L}(V,W)$ is a space of maps (Definition 2.4.1)	Chapter 24 §§08 and 11 each declare their use at the head of the section; the plain letters carry a subscript
$\mathbf{R}$ , $r$	the $\mathbf{R}$ of a $QR$ factorization; the plane rotation $\mathbf{R}_\theta$ (Theorem 11.7.6); the rotation through $2\pi/n$ in Chapter 23; the absorbing block of a Markov chain (Definition 19.6.1). Plain $R_{\mathbf{A}}(\mathbf{x})$ is the Rayleigh quotient (Definition 17.1.1) and $r_i(\mathbf{A})$ a Gershgorin radius (Definition 20.1.1)	the Rayleigh quotient is plain italic and takes a vector; the Gershgorin radius is plain and takes an index
$\mathbf{Q}$	the orthogonal factor of a $QR$ factorization; the orthogonal linear part of a Euclidean motion (Definition 22.4.1); the quarter turn in Chapter 23; the transient block of an absorbing chain (Definition 19.6.1). Blackboard $\mathbb{Q}$ is the rationals (Definition 0.3.1)	the chapter, and the blackboard face for the field
$\mathbf{S}$ , $\mathcal{S}$	a congruence or a similarity matrix; a Schur complement in Chapter 24 §02; the sample covariance (Definition 24.10.2); the cyclic shift in Chapter 12 §09. Calligraphic $\mathcal{S}_{\mathbf{A},\mathbf{B}}$ is the Sylvester operator (Definition 12.10.1)	each section of Chapter 24 says which it means; the Sylvester operator carries both its matrices
$\mathbf{N}$ , $\mathcal{N}$	the fundamental matrix $(\mathbf{I} - \mathbf{Q})^{-1}$ of an absorbing chain (Definition 19.6.1); an incidence matrix, with $\mathbf{L} = \mathbf{N}\mathbf{N}^{\top}$ (Definition 3.7.1); the strictly upper triangular matrices $\mathcal{N}$ (Example (Six subalgebras)). Blackboard $\mathbb{N}$ is the natural numbers	the calligraphic face for the algebra, the blackboard face for the number set
$\mathbf{J}$	the all-ones matrix (Example (Orthogonally diagonalizing a 3 by 3 matrix)); the Jordan block $\mathbf{J}_k(\lambda)$ (Definition 10.3.1)	the Jordan block always carries its size and its eigenvalue
$\mathbf{C}$	the companion matrix $\mathbf{C}(p)$ (Definition 10.5.3); the real Jordan block $\mathbf{C}_k(\lambda)$ (Definition 10.4.1); a circulant (Definition 12.9.1). Plain $C$ and $C^a_b$ are contractions (Definition 15.4.1) and $C_{ij}$ a cofactor (Definition 7.6.1); blackboard $\mathbb{C}$ is the complex numbers	the companion matrix always carries its polynomial, the Jordan blocks their size and eigenvalue; a bare $\mathbf{C}$ in Chapter 23 is a circulant
$\mathbf{W}$ , $W$	the Wielandt matrix $\mathbf{W}_n$ (Theorem 19.5.6); the unnormalized transform matrix $\mathbf{W}_n$ of Chapter 24 §12; the accumulated orthogonal factor $\mathbf{W}_k$ of Chapter 24 §08. Plain $W$ is a subspace, $W(\mathbf{A})$ the numerical range (Definition 20.2.1), and $W_n$ the zero-sum matrices of Chapter 19 §07 (Lemma 19.7.1)	the numerical range always carries its matrix as an argument; the subspace never carries one
$\Theta$ , $\theta$	the isomorphism $\mathbb{C} \to \mathbb{R}^2$ (Example (The Standard Isomorphisms)); the basis-dependent $V \to V^{}$ (Theorem 5.3.2); the natural isomorphism $V^{} \otimes W \to \mathcal{L}(V,W)$ (Theorem 15.3.9); the diagonal matrix of principal angles (Definition 20.8.1). Lowercase: the angle between vectors (Definition 11.1.5), the choice-of-origin map $\theta_O$ (Proposition 22.1.3), the Ritz values $\theta_1, \dots, \theta_k$ (Definition 24.6.2)	the principal-angle $\Theta$ appears only inside $\sin\Theta$ and $\cos\Theta$ ; lowercase $\theta$ is an angle unless the section says otherwise, and the choice-of-origin map carries the point $O$
$\Pi$ , $\pi$	bold $\boldsymbol{\Pi}$ is the limit of a convergent stochastic matrix (Theorem 10.10.4); plain $\Pi$ is a plane in Chapter 22 §§04, 08, 09 and the quotient map $\operatorname{GL}(V) \to \operatorname{PGL}(V)$ in §07; $\pi$ is the quotient map $X \to X/{\sim}$ of a set (Definition 0.5.5), a quotient homomorphism of algebras (Definition 23.2.2), and, locally, a slot projection	the typeface, and that no section of Chapter 22 uses both meanings
$\Phi$ , $\Psi$	a symmetric gauge function and its dual (Definition 21.3.1, Definition 21.3.3); the two isomorphisms of Chapter 23 §09 (Definition 23.9.5)	the argument: a gauge eats a real vector, Chapter 23’s maps eat an element of a group algebra
$\Omega$ , $\omega$	plain $\Omega_n$ is the doubly stochastic matrices (Definition 18.8.4), bold $\boldsymbol{\Omega}_{2m}$ the standard alternating matrix (Definition 14.10.2). Lowercase $\omega$ is a root of unity (Definition 19.5.2, Definition 24.12.1), an angular frequency (Definition 24.9.2), the exponent of matrix multiplication (Definition 24.13.2), and a differential form being wedged (Definition 15.10.1)	the typeface for $\Omega$ ; for $\omega$ , the section, each of which declares its use
$\rho$	spectral radius (Definition 16.4.1); a representation of a group or an algebra (Definition 23.7.1)	the argument: a matrix for the radius, a group element for the representation
$\sigma$	a permutation (Definition 7.3.2); a singular value $\sigma_i(\mathbf{A})$ (Definition 13.9.1)	the singular value always carries an index and a matrix
$\lambda$ , $\mu$	a scalar or an eigenvalue throughout; also the linear-part homomorphism $\operatorname{E}(n) \to \operatorname{O}(n)$ (Definition 22.4.1) and the Fiedler value $\mu_2$ (Definition 24.11.2)	Chapter 22 §04 is the one place $\lambda$ is not a scalar, and it says so
$\lvert \cdot \rvert$	modulus of a complex number (Definition 0.6.2); cardinality of a set; the matrix absolute value $(\mathbf{A}^{}\mathbf{A})^{1/2}$ (Definition 13.2.1); the entrywise* absolute value in Chapter 19 (Definition 19.1.1)	what sits inside. Chapter 19 states its override, and the Chapter 13 meaning does not appear there
$\Lambda$	plain $\Lambda^k V$ is an exterior power (Definition 15.8.2); bold $\boldsymbol{\Lambda}(\lambda)$ is a rotation-scaling block (Definition 10.4.1)	the typeface, and the block always carries its eigenvalue
$\delta$	Kronecker delta $\delta_{ij}$ (Definition 0.9.3); a point mass $\delta_g$ in a group algebra (Definition 23.1.5); a basis element $\delta_s$ of a free vector space (Definition 15.2.3); one rounding error (Definition 24.1.1)	the Kronecker delta always carries two numerical subscripts
$\perp$	orthogonal complement $U^{\perp}$ (Definition 11.3.1); the complement $U^{\perp_\beta}$ for a bilinear form (Definition 14.1.3); orthogonal direct sum $U \perp W$ of a form (Definition 14.7.3); a polarity $S^{\perp}$ in projective space (Theorem 22.8.3)	the unadorned $U^{\perp}$ is always the inner-product one
$\cong$ , $\sim$ , $\simeq$ , $\approx$	isomorphic spaces (Definition 2.5.3) or, between spaces carrying forms, isometric (Definition 14.8.1); similar matrices (Definition 4.2.2); congruent matrices (Definition 14.2.1); equivalent over $F[x]$ (Definition 10.7.2), or “approximately equal” in prose	what stands on either side; the four relation symbols are never interchanged
brackets	$[v]_{\mathcal{B}}$ a coordinate vector (Definition 1.5.2); $[T]_{\mathcal{B}}^{\mathcal{C}}$ a matrix of a map (Definition 4.1.1); $[\mathbf{A},\mathbf{B}]$ a commutator (Definition 8.5.1); $[x,y]$ a segment (Definition 18.1.1); $[\mathbf{x}]$ a projective point (Definition 22.6.2) and $[T]$ a projectivity (Definition 22.7.1); $F[x]$ and $F[G]$ rings and algebras (Definition 0.8.2, Definition 23.1.5)	the number of arguments and whether they are bold: a segment takes two bold vectors, a commutator two bold matrices
the superscript of $[\,\cdot\,]_{\mathcal{B}}^{\mathcal{C}}$	for a linear map it is the output basis, $[T]_{\mathcal{B}}^{\mathcal{C}}$ (Definition 4.1.1); for a bilinear form it is the basis read into the second slot, $[\beta]_{\mathcal{B}}^{\mathcal{B}}$ (Definition 14.1.2)	what sits inside the brackets. The two readings are the same rule — the basis on the second slot — since a map’s second slot is its output; a form on $V \times V$ always gives both slots the same basis, so its two labels agree, and neither is ever left off
$T$	a linear map throughout; the transpose $\mathbf{A}^{\top}$ ; Chebyshev polynomials $T_n$ (Definition 24.7.4); the tensor algebra $\operatorname{T}(V)$ (Definition 15.6.2) and mixed tensors $T^p_q(V)$ (Definition 15.4.2); a $T$ -transform (Definition 21.1.1)	the tensor algebra is upright and takes a space; the Chebyshev polynomial and the $T$ -transform carry subscripts
$P$ , $p$	orthogonal projection $P_U$ (Definition 11.3.2); Legendre polynomials $P_n$ (Definition 11.10.1); a point of an affine space (Definition 22.1.1); the nearest-point map $P_C$ (Theorem 18.3.1); a face $P_J$ (Proposition 22.3.1); the even-odd permutation $\mathbf{P}_n$ (Definition 24.12.2); characteristic polynomial $p_T$ (Definition 7.10.1); a gauge $p_K$ (Definition 18.4.1)	a subspace subscript means a projection, an integer subscript a polynomial; bold means a matrix
$Z$	the cyclic subspace $Z(v;T)$ (Definition 10.5.1); the center $Z(A)$ of an algebra (Definition 23.1.6); the zero set $Z(q)$ of a quadratic form (Definition 22.10.1); a Z-matrix (Definition 19.9.1)	the argument decides: a vector and an operator, an algebra, or a form
$\mathbf{E}$ , $E$	the perturbation in $\mathbf{A} + \mathbf{E}$ ; a matrix unit $\mathbf{E}_{ij}$ (Example (Standard Bases)); an elementary matrix (Definition 3.4.1); an eigenspace $E_\lambda(T)$ (Definition 9.2.2); the Euclidean group $\operatorname{E}(n)$ (Definition 22.4.1)	subscripts again: two indices give a matrix unit, an eigenvalue gives an eigenspace, a bare $\mathbf{E}$ is the perturbation
$\mathbf{D}$ , $D$	a diagonal matrix (Definition 0.9.4); the elementary scaling $\mathbf{D}_i(c)$ (Definition 3.4.1); the twiddle matrix $\mathbf{D}_m$ (Definition 24.12.2); the degree matrix in $\mathbf{L} = \mathbf{D} - \mathbf{A}$ (Definition 24.11.1); a Gershgorin disc $D_i(\mathbf{A})$ (Definition 20.1.1); density matrices $\mathcal{D}_n$ (Definition 18.8.3)	the Gershgorin disc is plain italic, since the bold letter is a diagonal matrix
$\mathbf{M}$ , $M$	a block matrix, as in the Schur complement $\mathbf{M}/\mathbf{A}$ (Definition 8.3.1); the mass matrix (Definition 24.9.2); the plane reflection $\mathbf{M}_\theta$ (Theorem 11.7.6); a minor $M_{ij}$ (Definition 7.6.1); the space $M_n(F)$ (Definition 0.9.1); an M-matrix (Definition 19.9.2)	plain italic with two indices is a minor; $M_n(F)$ always carries its size and field
$u$	the unit roundoff (Definition 24.1.1); bold $\mathbf{u}$ is a vector	the typeface; Chapter 24 §01 warns about the pair
$h$	the period of an irreducible nonnegative matrix (Definition 19.5.2); a support function $h_K$ (Definition 18.4.2); a mesh width in Chapter 19 §09	the support function always carries its convex set
simple	a simple algebra (Definition 23.3.1); a simple $A$ -space (Definition 23.3.3); a simple eigenvalue, of multiplicity one (Definition 20.11.1); a simple tensor (Definition 15.2.2)	what the word is applied to
equivalent	equivalent matrices $\mathbf{B} = \mathbf{Q}\mathbf{A}\mathbf{P}$ (Definition 4.3.1); equivalence over $F[x]$ (Definition 10.7.2); equivalent norms (Definition 16.2.1); equivalent representations (Definition 23.7.3); affine equivalence of quadrics (Definition 22.5.2)	the objects being compared
positive	$\mathbf{A} > \mathbf{O}$ entrywise, every entry positive (Definition 19.1.1); $\mathbf{A} \succ \mathbf{O}$ positive definite (Definition 13.1.1)	the symbol: $>$ is entrywise, $\succ$ is the Loewner order. A positive matrix need not be positive definite, nor conversely
radical, contraction	radical of a bilinear form (Definition 14.1.3) or of an algebra (Definition 23.3.5); contraction of tensor slots (Definition 15.4.1) or a matrix of norm at most one (Definition 21.8.1)	the argument, in both cases
signature	the pair $(n_+, n_-)$ , and also the single integer $n_+ - n_-$	Definition 14.5.1 says which is meant where the word appears