Special directions
The instrument feeds the matrix $A = \begin{pmatrix}2 & 1\\ 1 & 2\end{pmatrix}$ a ring of 24 unit vectors: each faint spoke is an input $\mathbf{v}$, each thin line is its output $A\mathbf{v}$. As the probe sweeps the full circle, how many directions keep $A\mathbf{v}$ on the same line as $\mathbf{v}$?
At $\theta = 20°$ the ∠ v, Av stat reads $16.2°$ — the matrix has knocked $\mathbf{v}$ off its line. At $90°$ the gap is $26.6°$. But at $45°$ it collapses to $0.0°$, both lines flash, and λ snaps to $3.00$. Sweep on: $135°$ fuses again with $\lambda = 1.00$, then $225°$ and $315°$ repeat the pair from the opposite side. Four probe angles, but only two lines through the origin — exactly the dashed rays.
You caught the matrix doing nothing but scaling. At $45°$ the output does not merely point near $\mathbf{v}$ — it sits on the very same ray, three times as long. A nonzero vector with that property is an eigenvector of $A$, and its scale factor is the eigenvalue $\lambda$:
The equation is geometric. The left side is the full machinery of matrix multiplication, both coordinates mixing into both outputs; the right side is scaling by one plain number. On an eigenvector the two agree exactly. The second survivor at $135°$ shows that survival needs no drama: there $\lambda = 1$, and the output lands exactly on top of the input. Every other direction fails the test — which is why the λ stat reads — almost everywhere.
At $\theta = 45°$ the stats read ∠ v, Av $= 0.0°$ and $\lambda = 3.00$. What did the matrix do to $\mathbf{v}$?
Switch to the shear preset $\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}$ and sweep the probe. How many invariant lines does the shear keep?
Transfer: $\mathbf{v}$ is an eigenvector of $A$ with $\lambda = 3$. What is $A(5\mathbf{v})$?
An eigenvector is really an invariant line: every nonzero multiple of it is an eigenvector with the same $\lambda$. These lines are the axes the transformation itself prefers — along them, all the machinery of a matrix collapses to multiplication by one number.