2×2 State-Space Eigenvalues
Eigenvalues of a 2×2 system matrix A — the poles that set stability, natural frequency, and damping.
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The engineering
The eigenvalues of the state matrix A are the poles of a linear time-invariant system — they alone determine the free-response modes. Real negative eigenvalues decay exponentially; a complex conjugate pair oscillates and decays; anything with a positive real part is unstable. For a 2×2 A the characteristic polynomial is just λ² − (tr A)λ + det A, so you never need a matrix routine.
Quick stability check without computing roots: a 2×2 system is stable exactly when tr A < 0 and det A > 0 (both eigenvalues in the left half-plane). When the roots are complex, ωₙ = √(det A) and ζ = −tr A / (2·ωₙ), which maps the poles straight onto the second-order damping picture. Watch the sign convention — controls texts write ẋ = Ax, so the poles are the eigenvalues of A, not −A.
Where this math comes from
The word "eigenvalue" (from German eigen, "own" or "characteristic") entered mathematics through David Hilbert's 1904 work on integral equations, but the underlying idea of a characteristic root traces back to Cauchy and Cayley in the 19th century. The characteristic equation det(A − λI) = 0 is a direct product of the matrix algebra Arthur Cayley formalized in 1858.
State-space control brought eigenvalues to the bench. Rudolf Kálmán reframed control theory around the state vector around 1960, and textbooks like Chi-Tsong Chen's Linear System Theory and Design made "the eigenvalues of A are the poles of the system" the first thing every controls student learns. For a hand-sized system the trace-determinant shortcut turns pole placement into mental arithmetic.
- 1858Arthur CayleyFormalizes matrix algebra and the characteristic equation.
- 1904David HilbertCoins 'eigenvalue' in his theory of integral equations.
- 1960Rudolf KálmánPuts the state vector at the center of modern control theory.
- 1984Chi-Tsong ChenLinear System Theory and Design codifies eigenvalues as system poles for a generation of engineers.
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