Lesson 00 · Prerequisites

Eigenvectors, eigenvalues and the Jacobian — with the right pictures

The two ideas from algebra that hold up the whole notebook: the "own" directions of a transformation, and its local linear imitation. Each concept comes with a famous metaphor to keep in your pocket.

Where they're needed — the covariance $\Sigma$ of lesson 01 is read through its eigenvalues (axes and semi-axes of the ellipsoid); the EWA projection of lesson 05 is a Jacobian; and the determinant tells you how much a splat's area inflates on screen.

Eigenvectors and eigenvalues: the rails of the transformation

A matrix $A$ transforms the entire plane in one stroke: every vector gets moved, rotated, stretched. The question that defines eigenvectors is: is there any direction the transformation respects? That is, a vector $\mathbf v$ that doesn't get deviated, but only slides along its own line:

$$A\,\mathbf v \;=\; \lambda\,\mathbf v$$

Two objects, two different questions — this is the point that often gets muddled:

Mnemonic — the eigenvector answers "which rail stays a rail?" (a direction). The eigenvalue answers "how fast does the train run on that rail?" (a number: the stretch factor).

Three famous pictures

1 · The Earth's axis. The Earth's rotation moves everything: Rome travels, New York travels, the whole ocean travels. The only direction that goes nowhere is the axis: the poles stay put. The axis is the eigenvector of the rotation matrix, with eigenvalue $\lambda = 1$ (it doesn't even stretch). It's a theorem by Euler: every 3D rotation has an axis — translated: every 3D rotation matrix has a real eigenvector with $\lambda=1$. Not so in 2D: a rotation of the plane leaves no direction fixed (see below).

2 · Dough under the rolling pin. Roll the dough out in one direction: it stretches ×2 along the rolling pin and shrinks ×0.7 in the perpendicular one. If you had drawn lots of little arrows on the dough beforehand, almost all of them rotate as you roll (they get dragged toward the stretch direction); the only ones that don't rotate are the one along the rolling pin and the perpendicular one. Those two directions are the eigenvectors; 2 and 0.7 are the eigenvalues.

3 · The sheared Mona Lisa. The most famous figure on Wikipedia about this topic: apply a shear to the painting and everything tilts — except the vector along the base, which stays exactly where it was ($\lambda=1$). Fun fact: in a shear there is only one rail (a "defective" matrix: two equal eigenvalues, a single independent eigenvector) — try the preset below.

Eigen in German means "own, characteristic": eigenvectors are the transformation's own directions, the ones that belong to it.

The λ bestiary

Eigenvalue	What it does to its rail	Picture
$\lambda > 1$	stretches	the rolling pin
$0 < \lambda < 1$	compresses	the spring shortening
$\lambda < 0$	flips the direction (the line stays put)	the rubber band snapping to the other side
$\lambda = 0$	annihilates the direction (everything to the origin)	the steamroller — and indeed $\det A = 0$, not invertible
complex $\alpha \pm \beta i$	no real rail: everything rotates	the 2D rotation — every direction turns

a 1.60 b 0.00 c 0.00 d 0.50 apply t 1.00

Fig. 1 — interactive. The matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ applied to the plane: unit circle → ellipse (blue), unit square → parallelogram with the "F" showing its orientation, terracotta arrows = 12 generic directions, green arrows and lines = the eigenvectors (the "rails"). Drag apply t back and forth: all the arrows swerve except the ones on the rails, which merely stretch by λ. With "Rotation" the rails vanish (complex eigenvalues); with "Reflection" the F mirrors itself (det < 0).

In 3DGS you live inside this — the factorization $\Sigma = R\,S\,S^{\!\top}\!R^{\!\top}$ of lesson 01 is literally an eigendecomposition: the columns of $R$ are the eigenvectors of $\Sigma$ (the ellipsoid's axes), the squared scales $s^2$ are the eigenvalues (the variances along the axes). The "Covariance" preset builds $A$ exactly this way, with axes at 30° and scales $(1.5,\,0.6)$ — and indeed you read $\lambda_1 = 2.25 = 1.5^2$, $\lambda_2 = 0.36 = 0.6^2$. When the optimizer updates a Gaussian's scales it is doing gradient descent on the eigenvalues of its covariance.

A bonus to cement the "eigen" family: PCA takes the eigenvectors of the covariance of a swarm of points (the axes of the swarm); PageRank is the dominant eigenvector of the web graph — Google built an empire on it; a crystal glass shatters at its note because it has eigenfrequencies. Every time you hear "eigen-something", the underlying question is always the same: what stays itself under this operation?

The Jacobian: the local flat map

One-line refresher: the derivative is a zoom. A smooth function, magnified enough around a point, becomes indistinguishable from its tangent line. The Jacobian is the exact same idea for $f:\mathbb R^n \to \mathbb R^m$: zoom in enough on the deformation around a point $\mathbf p$ and it becomes indistinguishable from a linear map. That linear map is $J(\mathbf p)$:

$$J_{ij} \;=\; \frac{\partial f_i}{\partial x_j} \qquad\qquad f(\mathbf p + \boldsymbol\delta) \;\approx\; f(\mathbf p) + J(\mathbf p)\,\boldsymbol\delta$$

Row $i$ lists how strongly output $i$ reacts to each input: the Jacobian is the sensitivity table of $f$ at that point. But the pictures that stick are these two:

1 · The neighborhood map. Projecting the (curved) Earth onto a (flat) sheet is a nonlinear map — the perfect world map doesn't exist. Yet the map of your neighborhood works just fine: at small scale the projection is approximated beautifully by a linear map. Every point on the planet has its own "local flat map", and that map is the Jacobian of the projection at that point.

2 · The funhouse mirror. Your full figure is monstrous: short legs, pear-shaped head — a decidedly nonlinear transformation. But a postage stamp of skin, a mole and its surroundings, is simply stretched, rotated and sheared: an affine transformation. The funhouse mirror is a collection of Jacobians, one for each point of your silhouette.

Why we need it — perspective projection divides by depth: nonlinear. And a Gaussian pushed through a nonlinear map is no longer Gaussian (the shells bend). The EWA trick (lesson 05): near the center $\mu$ replace the projection with its Jacobian — a linear map takes Gaussians to Gaussians — and you get $\Sigma' = J\,W\,\Sigma\,W^{\!\top}\!J^{\!\top}$. The radial distortion of lenses (the one COLMAP estimates when it calibrates cameras) is a nonlinear warp too: the demo below uses exactly that.

strength 0.70 probe radius 0.55

Fig. 2 — interactive. A nonlinear map warps the grid. The little circles become ellipses — this is exactly the Tissot indicatrix from atlases — colored by $\det J$: terracotta where area inflates, blue where it compresses. Drag the probe: the solid circle is the true image of the circle, the dashed one is the Jacobian's linear prediction. Now shrink the probe radius: the two curves coincide — "zooming until it becomes linear" is the whole essence of the Jacobian. Negative strength = barrel (GoPro), positive = pincushion.

The determinant: how much areas inflate

For a linear map the determinant is the area factor (in 3D: the volume factor): take the unit square, apply $A$, you get a parallelogram; $|\det A|$ is its area. Rolling pin ×2 in one direction and ×1.5 in the other → every little patch of area ×3. In Fig. 1 it is the parallelogram with the "F" inside, and its value is in the readout.

With the rails picture the fact that $\det A = \lambda_1 \lambda_2 \cdots$ becomes obvious: you stretch ×$\lambda_1$ along one rail and ×$\lambda_2$ along the other, so the area goes ×$\lambda_1\lambda_2$. And the two pathological cases have a precise physics:

Case	Picture	Meaning
$\det = 0$	the steamroller: it squashes the plane onto a line (one eigenvalue is 0)	not invertible — you can't un-squash a crêpe: two points that ended up in the same place can no longer be told apart
$\det < 0$	the glove turned inside out: the right hand in the mirror becomes a left one	the area is still there ($\|\det\|$) but the orientation has flipped — in Fig. 1 the "F" mirrors itself with the "Reflection" preset

The determinant of the Jacobian: local area factor

If the Jacobian is the local linear map, its determinant is the local area factor — a different number at every point, telling you how much the map inflates or compresses the little areas there. Three famous anchors:

1 · Mercator's Greenland. On the Mercator world map Greenland looks as big as Africa; in reality it is about 14 times smaller. The reason is that the $\det J$ of the Mercator projection blows up approaching the poles. The Tissot ellipses printed in atlases are exactly Fig. 2 applied to the world map: equal little circles on the sphere, ever more inflated ellipses on the chart.

2 · The pizza slice. In the change to polar coordinates, $dx\,dy = r\,dr\,d\theta$: that $r$ is $|\det J|$. The picture: a "little square" $\Delta r \times \Delta\theta$ near the edge of the pizza is much bigger than the same little square near the center — the area depends on where you are, i.e. on $r$. Every time you change variables in an integral, $\det J$ is there acting as the exchange rate between areas.

3 · Honey on the dough (for the ML part of your brain). Spread a film of honey on the dough and then roll it out: where the dough widens, the film thins exactly by the area factor — mass is conserved. It's the change of variables for probability densities, $p_Y(\mathbf y) = p_X(\mathbf x)\,/\,|\det J|$: the heart of normalizing flows, and the reason a linearly transformed Gaussian stays normalized with the right covariance.

In 3DGS — the footprint of a splat on screen is the ellipse of the projected covariance $\Sigma'$, and its area scales as $\sqrt{\det \Sigma'}$. Anti-aliasing tricks (e.g. Mip-Splatting) correct the opacity precisely with a ratio of determinants, to keep splats that go sub-pixel from "vanishing". The determinant is not a technicality: it is the renderer's bookkeeping of areas.

Fridge-door summary

Object	Question it answers	Picture	Where you'll see it again
eigenvector	which direction doesn't get deviated?	the Earth's axis; the rail	ellipsoid axes = columns of $R$
eigenvalue	by how much does it stretch along that rail?	the rolling pin ×2	variances = squared scales $s^2$
Jacobian	how does it stretch space here?	the neighborhood map; the funhouse mirror	EWA projection, $\Sigma' = JW\Sigma W^{\!\top}\!J^{\!\top}$ (lesson 05)
det (of the Jacobian)	how much do areas inflate here?	Mercator's Greenland; the pizza slice	splat area, changes of variable, anti-aliasing

Eigenvalue	What it does to its rail	Picture
\(\lambda > 1\)	stretches	the rolling pin
\(0 < \lambda < 1\)	compresses	the spring shortening
\(\lambda < 0\)	flips the direction (the line stays put)	the rubber band snapping to the other side
\(\lambda = 0\)	annihilates the direction (everything to the origin)	the steamroller — and indeed \(\det A = 0\), not invertible
complex \(\alpha \pm \beta i\)	no real rail: everything rotates	the 2D rotation — every direction turns

Case	Picture	Meaning
\(\det = 0\)	the steamroller: it squashes the plane onto a line (one eigenvalue is 0)	not invertible — you can't un-squash a crêpe: two points that ended up in the same place can no longer be told apart
\(\det < 0\)	the glove turned inside out: the right hand in the mirror becomes a left one	the area is still there (\(\|\det\|\)) but the orientation has flipped — in Fig. 1 the "F" mirrors itself with the "Reflection" preset