Notation

symbol	meaning
$x_i$ , $w_i$	article column vector and its covariance fit weight
$\mu$ , $\Sigma$ , $Z$	weighted article mean, covariance, and total fit weight
$m$ , $M$ , $W$	weighted vector sum, outer-product sum, and diagonal fit-weight matrix
$V$ , $\Lambda$	eigenvectors (columns) and eigenvalues of $\Sigma$ ; $k$ of them kept
$R$ , $V' = VR$ , $y=R^{\top}c$	naming rotation, named directions, and named coordinate column
$c$ , $\hat{x}$ , $r$	raw coordinates, reconstruction, residual of an observation column $x$
$\Gamma=R^{\top}\Lambda R$	covariance of the rotated raw coordinates; generally not diagonal
$T^2$ , $Q$ , $\nu$	Hotelling’s statistic, residual energy, novelty ratio
$T$ , $\mu_T$ , $S$ , $\beta$	TF‑IDF rows, their naming mean, whitened raw score rows, centred term–score loadings
$U$ , $\Sigma_{\mathrm{svd}}$ , $V$	left directions, singular-value matrix, and right directions in the SVD context
$\sigma_j$	singular values; $\lambda_j=\sigma_j^2/n$ for an equally weighted centred matrix
$\varepsilon_s$ , $s_j$	story attention energy; empirical named-axis standard deviation for dominance
$n_\tau$ , $w_i = 1/n_{\tau(i)}$	articles on day $\tau$ and the day-balanced weight of article $i$
$T_b$ , $S_b$ , $N_b$	term rows, score rows, and article count in block $b$
$S_1, S_2, S_3$	power sums of an axis’s scores, from which its skewness sign is read

The main symbols above keep their stated meaning throughout the pipeline. The following are local calculation symbols; each is defined again where it is used. A shared letter in two explicitly separated examples does not assert that the objects are equal.

Local symbols	Meaning and scope
$e,\pi$	Natural-logarithm base and circle constant; one full turn is $2\pi$ radians.
$a_t,b_t$ ; $\mathrm{df},\mathrm{tf},\mathrm{idf}$	Entries of two term vectors; document frequency, damped term frequency, and inverse document frequency in Chapter 2.
$A,B,C$ ; $A_{ij}$	Example matrices and an entry selected by row $i$ and column $j$ in Chapter 3.
$X_c$ , $\mathbf1$ , $I$	Centred data table; a column of ones of the required row count; identity matrix of the stated square size.
$u,v,v_j,\alpha_j$	Candidate direction, eigenvector, eigenvector number $j$ , and coefficient along it in the covariance chapter.
$a_i$	Scalar projection score of centred observation $i$ along a temporary unit direction in the eigenvalue proof.
$\lambda_+$ , $\sigma^2$	Model noise edge and stipulated noise variance in the Marchenko–Pastur illustration.
$q$ , $U_k,V_k,\Sigma_k$	Smaller data-table dimension; truncated factors retaining $k$ directions in Chapter 5.
$X_k$	Rank-at-most- $k$ reconstructed table in the truncation chapter.
$\theta,J,h$	Proposed scalar fit, its squared-error function, and a nonzero change in the fit in the differentiation lesson.
$J'(\theta),\,dJ/d\theta$	Derivative of the scalar error function. The prime means differentiation here, unlike the named-basis prime later.
$l,\Omega,Y,Q_{\mathrm{range}}$	Working width, random input table, output sketch, and orthonormal working basis for randomized SVD.
$q_1,q_2$	Unit columns built in the Gram–Schmidt teaching example.
$\ell_{ij},\phi$	Rotated loading entry and a pairwise rotation angle in radians in the varimax calculation.
$x,y,u_i,v_i$	Two local loading columns and rowwise squared-loading differences and products in the pairwise varimax formula.
$A,B,C,D,C_,D_$	Four scalar sums and two centred combinations in that local formula.
$R(\phi)$	Two-dimensional rotation through the angle $\phi$ .
$y_s,y_{sj},\bar y_{\mathrm{day}}$	Story $s$ ’s named column, its entry on axis $j$ , and a day’s attention-weighted spectrum.
$a_1,a_2$	Top and runner-up scores in the explicitly illustrative mixed-label heuristic.
$a,h_{\mathrm{old}},h_{\mathrm{new}},\delta$	Incoming value, running means, and departure from the old mean in Welford’s update.
$A_{\mathrm{old}},A_{\mathrm{new}}$	Old and new squared-deviation sums in that update.
$E,\varepsilon,g,\theta$	Covariance perturbation, its operator norm, the specified eigenvalue separation, and the turning angle in the stability bound. Here $\theta$ is an angle, unlike the scalar fit in Chapter 5.
$g(x_i)$	A fixed per-row contribution function in the block-sum identity; unrelated to the eigengap scalar.
$\mu_{T,w}$	The term-vector mean under day-balanced fit weights, distinguished from the unweighted naming mean $\mu_T$ .
$c_{ij},\bar c_j,m_2,m_3$	Raw score of article $i$ on axis $j$ , its unweighted orientation-pass mean, and second/third centred moments.

For operators, $\mathrm{diag}$ creates a diagonal table; $\min$ chooses the smallest value; $\max$ chooses the largest; and $\arg\max$ returns the index attaining the largest. A superscript $-1$ means a reciprocal for a nonzero scalar and an inverse for an invertible square matrix. Fractional powers of positive scalars express roots. The trigonometric functions $\sin$ and $\cos$ describe unit-circle components; $\operatorname{atan2}$ reverses that description while keeping the quadrant. The subscript $F$ on a matrix norm means Frobenius norm; subscript $2$ in the stability bound means Euclidean operator norm. Chapter 5 explains the limit used to define a derivative. Chapter 10 explains the rate bound denoted by big-O notation.