Each named scenario fixes four original points. Centering subtracts each coordinate’s mean and does not scale feature units. The plot uses centered originals and centered reconstructions; the exact table reconstructs raw coordinates by adding the mean. The sample covariance divides centered product sums by 3. Its principal eigenvalues are the variances along principal directions, sorted largest first.

For symmetric covariance [[a,b],[b,d]], principal variances are (a+d ± sqrt((a−d)²+4b²))/2. When they differ, the leading line angle is half atan2(2b,a−d), modulo 180°. Reversing its sign changes score signs while preserving reconstruction. Equal positive eigenvalues make every first direction a maximum, so there is no unique PC1 line. A zero covariance cloud also has no unique direction and makes variance ratios undefined.

Manual axes u and v are orthonormal; their score variances sum to total variance. Keeping one score drops the other variance; keeping both spans the full 2D space. Mean Error² divides squared Euclidean residuals by 4 points; discarded sample variance uses divisor 3 and equals (4/3)Mean Error². Exact equivalent special-angle matrices and a complete-basis identity avoid artificial floating residuals. Other angles use trigonometry; rounded values do not determine zero denominators or identity.

Retained variance is geometry, not a probability, accuracy or assurance of downstream value. Scope excludes general-dimensional solvers, SVD derivation, standardization, whitening, probabilistic PCA, nonlinear embeddings, train/test pipelines, labels and automatic component-count selection. Actual numeric/scenario/count edits clear stale answers; prediction changes and Reset restore the current step.