Differential Equations Decoded 50 Repeated Eigenvalues and Generalized Eigenvectors

Free to reuse. Free to remix. No attribution required. Make your own at https://www.patreon.com/cw/MadSciHub QUICK SUMMARY Some 2x2 matrices have a repeated eigenvalue with only one eigenvector, and the old recipe produces one basis solution when we need two. We fix it with a generalized eigenvector w that satisfies (A - lambda I) w = v, giving the second basis solution (t v + w) e^(lambda t). This episode hands you a four-step recipe, a full worked example, the phase portrait (a degenerate node), and the exam traps your professor will use. KEY CONCEPTS 1. Defective Matrix - a matrix whose repeated eigenvalue has fewer independent eigenvectors than its algebraic multiplicity 2. Generalized Eigenvector - a vector w satisfying (A - lambda I) w = v, one step up the kernel chain from an eigenvector 3. Second Basis Solution - the form (t v + w) e^(lambda t) that completes the general solution of a defective 2x2 system 4. Degenerate Node - the phase portrait where every trajectory is tangent to the single eigenvector direction at both ends DEFINITIONS - Algebraic Multiplicity: how many times an eigenvalue appears as a root of the characteristic polynomial - Geometric Multiplicity: the dimension of the eigenspace for a given eigenvalue - Defective Matrix: a matrix with geometric multiplicity strictly less than algebraic multiplicity - Generalized Eigenvector of Rank 2: a vector w with (A - lambda I)^2 w = 0 but (A - lambda I) w nonzero - Degenerate Node: a defective repeated-eigenvalue phase portrait with one attracting or repelling direction - Jordan Chain: a sequence of generalized eigenvectors built from an eigenvector via pullbacks through (A - lambda I) HOW IT WORKS 1. Compute the characteristic polynomial det(A - lambda I) and find its roots 2. Row-reduce (A - lambda I) and count pivots to diagnose defective (one pivot) or non-defective (zero pivots) 3. Find the single eigenvector v from (A - lambda I) v = 0 4. Solve the linear system (A - lambda I) w = v for any particular generalized eigenvector w 5. Assemble the general solution X(t) = c_1 v e^(lambda t) + c_2 (t v + w) e^(lambda t) 6. Apply the initial condition X(0) = X_0 by solving c_1 v + c_2 w = X_0 for the two constants 7. Verify by differentiating and plugging back into X prime equals A X 8. Sketch the degenerate node phase portrait with trajectories tangent to v at both ends KEY ARGUMENTS 1. For a 2x2 system with repeated eigenvalue, only two cases exist — two eigenvectors (non-defective) or one (defective) 2. The naive guess X_2 = t v e^(lambda t) fails because coefficient matching forces v = 0, contradicting v being an eigenvector 3. Adding a new vector w to the ansatz, X_2 = (t v + w) e^(lambda t), absorbs the leftover v when the algebra balances 4. Coefficient matching forces exactly one equation: (A - lambda I) w = v, a standard linear system 5. This w is a generalized eigenvector of rank 2, rigorously defined, not a hack 6. The phase portrait is a degenerate node — every trajectory aligns with the single eigenvector direction at both time extremes 7. The same multiply-by-t pattern appears in scalar repeated roots (Episode 24), resonance, and neural network training plateaus KEY TAKEAWAYS - Memorize the shape (t v + w) e^(lambda t) — t multiplies v, w stands alone - The defining equation is (A - lambda I) w = v, right-hand side is v, never zero - Always verify your w by computing (A - lambda I) w and confirming it equals v - Different students can get different valid w values; w is unique only up to adding multiples of v - Real-repeated eigenvalues do not produce spirals or centers — stop before reaching for complex formulas - The defective case is ~90 percent of exam questions on repeated eigenvalues MEMORY HOOKS - A generalized eigenvector is someone who does not wear the uniform but whose best friend does - The defective case is a basketball team with four players and a folding chair — we call up a ringer named w - Multiply by t is nature's signature wherever two degrees of freedom try to occupy one slot SOURCE https://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx #DifferentialEquations #LinearAlgebra #Eigenvalues #Eigenvectors #GeneralizedEigenvector #JordanForm #PhasePortrait #ODE #MathEducation #ExamPrep #Decoded #madscilecture #decoded #differentialequations #math #science