The stretch factor

At $t = 0.70$ the stat reads $3.00$. At $t = 0.25$: still $3.00$. At $t = -0.80$ both vectors swing to the opposite side of the origin — and the stat still reads $3.00$. Press λ2 dir: the ratio pins at $1.00$ and the dot band vanishes, because $A\mathbf{v}$ lands exactly on $\mathbf{v}$. Press rotation: λ reads complex, and the dashed gap between the two tips never closes at any $t$.

The ratio cannot move because of linearity: $A(t\mathbf{u}) = t(A\mathbf{u}) = \lambda(t\mathbf{u})$. The scalar $t$ passes straight through the matrix, so the stretch factor belongs to the direction, not to any particular vector on it — and a length-$2$ vector maps to length $6$. To find the factors without a slider, rearrange $A\mathbf{v} = \lambda\mathbf{v}$ into $(A - \lambda I)\mathbf{v} = \mathbf{0}$: the matrix $A - \lambda I$ must flatten some nonzero vector, so its determinant must vanish. For a $2\times 2$ matrix that condition is a quadratic, the characteristic polynomial:

The default matrix has trace $4$ and determinant $3$, so $\lambda^2 - 4\lambda + 3 = (\lambda - 1)(\lambda - 3) = 0$ — the $3$ and $1$ printed on the rays. A rotation has trace $2\cos\theta$ and determinant $1$, so its discriminant $4\cos^2\theta - 4$ is negative for any genuine turn: the roots are complex, and no real line survives. The algebra agrees with the gap that never closed.