Solving Linear Systems

Differential Equations · Axiom Academy

LESSON Eigenvalue Method for Linear Systems Solve completely: eigenvalues and eigenvectors give every solution, in all three cases you'll ever meet. For any system x′ = Ax , the guess x = v e λt turns the differential equation into an algebra problem: 2. Case 1 — Real, Distinct Eigenvalues When has two different real roots , each gets its own real eigenvector, and each pair gives an honest exponential solution along a fixed direction. 3. Case 2 — Complex Conjugate Eigenvalues When the characteristic polynomial has no real roots, its roots come in a conjugate pair — and so does the eigenvector, . A single complex eigenpair still hands us two independent real solutions, via Euler's formula. 4. Case 3 — Repeated Eigenvalues When has a double root λ, there's only one genuine eigenvalue — but a 2×2 system still needs two independent solutions. If A isn't secretly diagonal ( ), there is only one eigenvector direction, so we manufacture a second solution with a generalized eigenvector . One method, three shapes: real eigenvalues stretch/shrink along fixed directions, complex eigenvalues spiral, and a repeated eigenvalue needs a generalized eigenvector to pick up the missing t -term. Scroll up to revisit any case.

This is the written version of the interactive lesson above. See the full Differential Equations course.