SAME EIGENVALUES

What stays the same

These continuous linear systems satisfy ẋ = Ax, with A = [[−1, k], [0, −2]]. The triangular matrix always has eigenvalues −1 and −2. The initial vector is always (0, 1). Both coordinates use the same fixed dimensionless units, and distance always means the Euclidean norm √(x₁² + x₂²).

State-space units are equal horizontally and vertically. Neither plot rescales with k or time. The time unit is fixed. The faint complete curves extend to t = 6; the formulas continue to zero as t → ∞.

What changes

Only the off-diagonal coupling k varies, from 0 to 12. The matrix norm and eigenvector geometry are not fixed. This is a different system in the same coordinates, not a change of units applied to one system.

For k ≠ 0 the matrix is nonnormal: AAᵀ ≠ AᵀA. Its stable eigenvalues establish asymptotic decay, but do not determine the norm at every earlier time. Nonnormality alone does not guarantee a transient rise for every k or initial vector.

Exact motion, no integrator

x₁(t) = k(e⁻ᵗ − e⁻²ᵗ)
x₂(t) = e⁻²ᵗ

Each displayed state is evaluated directly from this solution. Curves are sampled for drawing; time scrubbing does not accumulate integration error. At t = 0 the norm initially decreases for every k. The stronger cases then grow before their eventual decay.

The norm is a geometric distance. This exhibit does not assert a mechanical energy interpretation or demonstrate an unstable system.

Which peak?

The readout maximizes ‖x(t)‖₂ / ‖x(0)‖₂ for the single stated initial vector over all t ≥ 0. It checks t = 0 and every positive stationary time analytically; the limit at infinity is zero. The maximum can remain 1 at t = 0.

This differs from optimizing over all initial directions, which would require the operator norm of eᴬᵗ. Context: Oxford’s introduction to pseudospectra and nonnormal behavior.