Appendix F — Notation Reference

This appendix is the single place to look up any symbol used in this book. The conventions here are frozen: they are identical in Chapter 1 and in Chapter 40, and every chapter author was held to them. If a symbol on the page ever looks unfamiliar, it is defined here.

The guiding principle is simple and worth stating up front, because it makes the whole table predictable: the typeface tells you the kind of object. A plain italic lowercase letter is a number; a bold lowercase letter is a vector; an italic capital letter is a matrix. Once you internalize that, you can read most expressions before you have read their definitions — you will know that $A\mathbf{x}$ is "a matrix acting on a vector" purely from the fonts.

F.1 Scalars, vectors, and matrices

A note you will need by Chapter 1. Mathematics indexes from one: the first component of $\mathbf{v}$ is $v_1$, and the top-left entry of $A$ is $a_{11}$. But numpy indexes from zero: that same first component is v[0], and that same top-left entry is A[0, 0]. This is the single most common source of off-by-one confusion when you check a hand computation against code. Every chapter flags it the first time it bites; here is the rule once, plainly: the math symbol $a_{ij}$ corresponds to the Python expression A[i-1, j-1].

Vectors are columns unless we say otherwise. Writing $\mathbf{v}=(v_1,v_2,v_3)$ on a line is shorthand for the column

$$\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix},$$

and the row version is its transpose, $\mathbf{v}^{\mathsf{T}} = [\,v_1\ v_2\ v_3\,]$. The distinction matters the moment you multiply: $\mathbf{v}^{\mathsf{T}}\mathbf{w}$ is a scalar (the dot product), while $\mathbf{v}\mathbf{w}^{\mathsf{T}}$ is a matrix (an outer product).

F.2 Operations on matrices

A few conventions inside this group are worth stating explicitly, because they are easy to get subtly wrong:

• Always $A^{\mathsf{T}}$, never $A^{T}$. The transpose superscript is a sans-serif "T" (typeset \mathsf{T}). This is a deliberate choice so the transpose can never be confused with a variable named $T$ — which in this book is reserved for a linear transformation (see §F.6).
• $A^{*}$ is the conjugate transpose, also called the Hermitian transpose or adjoint. For a real matrix it is identical to $A^{\mathsf{T}}$; the conjugation only matters over $\mathbb{C}$. We use $A^{*}$ (rather than $A^{\mathsf{H}}$ or $A^{\dagger}$) throughout, including in the quantum-computing material of Chapters 21, 27, and 34.
• Operator names are roman, variables are italic. Compare $\det(A)$ (the operator "det" upright, the matrix $A$ italic) with the eigenvalue $\lambda$ (a variable, italic). This is why we write $\operatorname{rank}(A)$ and $\operatorname{tr}(A)$, not $rank(A)$ — the upright type signals "this is a named function, not a product of letters."

Common Pitfall. $(AB)^{\mathsf{T}} = B^{\mathsf{T}}A^{\mathsf{T}}$ and $(AB)^{-1} = B^{-1}A^{-1}$ — the order reverses. The notation reminds you why: transpose and inverse both "undo from the outside in," like taking off shoes then socks in the reverse order you put them on. Writing $A^{\mathsf{T}}B^{\mathsf{T}}$ for $(AB)^{\mathsf{T}}$ is one of the most common algebra slips.

F.3 Spaces, subspaces, and the four fundamental subspaces

The four fundamental subspaces have locked symbols in this book — they look the same every time, by design, because they are the organizing framework of Part III and recur everywhere after it. Read them as a matched set:

$$\underbrace{C(A)}_{\text{column space}} \quad \underbrace{N(A)}_{\text{null space}} \quad \underbrace{C(A^{\mathsf{T}})}_{\text{row space}} \quad \underbrace{N(A^{\mathsf{T}})}_{\text{left null space}}.$$

The notation itself encodes a theorem: the row space is written $C(A^{\mathsf{T}})$ because it is the column space of the transpose, and the left null space is $N(A^{\mathsf{T}})$ because it is the null space of the transpose. There is no separate "R(A)" or "LN(A)" to memorize — the symbols are built from $C(\cdot)$, $N(\cdot)$, and $A^{\mathsf{T}}$. (Strang writes these the same way; see Appendix H.)

F.4 Inner products, norms, and angle

Two conventions here repay attention:

• Norms use double bars: $\lVert \mathbf{v}\rVert$, never $|v|$ or ||v||. Single bars $|c|$ are reserved for the absolute value (or modulus) of a scalar; double bars are for the length of a vector. The default norm is the Euclidean one, $\lVert\mathbf{v}\rVert = \sqrt{\mathbf{v}\cdot\mathbf{v}}$, unless a subscript says otherwise (e.g., $\lVert\mathbf{v}\rVert_1$ for the sum of absolute values).
• $\mathbf{u}\cdot\mathbf{v}$ versus $\langle\mathbf{u},\mathbf{v}\rangle$. The dot notation is concrete — it means $\sum_i u_i v_i$ in $\mathbb{R}^n$. The angle-bracket notation is abstract: it stands for whatever inner product a given space is equipped with, including the integral inner product $\langle f,g\rangle=\int f g\,dx$ on function spaces (Chapters 22 and 34) and the Hermitian inner product on $\mathbb{C}^n$. When you see angle brackets, expect generality; when you see a dot, expect coordinates.

F.5 Eigenvalues and the canonical factorizations

These two factorizations are the climaxes of the book, and their letters carry fixed meanings:

• In the eigendecomposition $A = PDP^{-1}$: $D$ is the diagonal matrix of eigenvalues, and the columns of $P$ are the corresponding eigenvectors. Reading it left to right is reading a transformation: $P^{-1}$ changes coordinates into the eigenbasis, $D$ scales along each eigen-axis, and $P$ changes back. (When $A$ is symmetric, the spectral theorem of Chapter 27 lets us replace $P$ by an orthogonal matrix $Q$, giving $A = QDQ^{\mathsf{T}}$.)
• In the SVD $A = U\Sigma V^{\mathsf{T}}$: $U$ and $V$ are orthogonal (their columns are orthonormal), and $\Sigma$ is a diagonal matrix of nonnegative singular values $\sigma_1 \ge \sigma_2 \ge \dots \ge 0$. The capital sigma $\Sigma$ here is a matrix, not a summation sign — context (and the fact that it sits in a product of matrices) tells them apart. Geometrically the SVD reads right to left as rotate–stretch–rotate.

The Key Insight. Every symbol in $A=PDP^{-1}$ and $A=U\Sigma V^{\mathsf{T}}$ has a geometric job. $D$ and $\Sigma$ do the stretching; $P$, $U$, and $V$ do the reorienting. The notation is not bookkeeping — it is a sentence describing what the matrix does to space.

F.6 Linear maps, and a few recurring conventions

A handful of typographic rules tie the whole system together:

• Inline math uses single dollars $...$; displayed equations use double dollars $$...$$. Anything mathematical — even a lone symbol like $\lambda$ in a sentence — is wrapped in math mode, never typed as a bare letter.
• A capital $T$ standing alone is a transformation; a capital $T$ in the superscript $A^{\mathsf{T}}$ is "transpose." They never collide because the transpose is always a raised sans-serif superscript attached to a matrix.
• Greek lowercase letters are scalars by default — $\lambda$ and $\mu$ for eigenvalues, $\sigma$ for a singular value, $\theta$ and $\phi$ for angles, $\alpha$ and $\beta$ for generic coefficients. The one exception is capital $\Sigma$ in the SVD (a matrix) and capital $\Lambda$, occasionally used for the diagonal eigenvalue matrix.
• Decorations are consistent: a hat $\hat{\mathbf{v}}$ marks a unit vector ($\lVert\hat{\mathbf{v}}\rVert=1$) or a fitted quantity such as $\hat{\mathbf{x}}$ in the regression of Chapter 17; a bar $\bar{x}$ marks a mean; a prime $\mathbf{e}_1'$ marks "where $\mathbf{e}_1$ goes after a transformation."

If you ever forget a symbol mid-chapter, this appendix is the fastest reference. The deeper rule never changes: the typeface names the object. Number, vector, matrix, transformation — the page tells you which one before you have read a single word.