MZB127 Chap.6 Linear ODE Systems and Eigenvalues
Linear ODE Systems and Eigenvalues
Coupled linear first-order equations are written compactly as a matrix system. Eigenvectors supply state-space directions and eigenvalues supply exponential time factors, so modal combinations build the general solution. The real parts of eigenvalues determine stability, imaginary parts create oscillation, and repeated eigenvalues require an independence check.
Phase portraits visualise state against state rather than state against time. Writing the state variables in a fixed order is essential because the same order defines matrix rows, vector components and plotted axes. The determinant equation identifies eigenvalues, while each null-space calculation identifies the associated direction.
Distinct real eigenpairs produce independent exponential modes whose coefficients are determined from the initial state. A positive real part creates expansion, a negative real part creates contraction and a mixture produces saddle behaviour. Complex conjugates generate rotation or oscillation with an envelope controlled by their shared real part.
A repeated eigenvalue may still have enough eigenvectors, so defectiveness must be tested rather than assumed. Direct multiplication verifies each eigenpair, and differentiating the assembled modal solution verifies the original system. The algebra and the phase portrait should tell the same forward-time story.
What this chapter covers
- 01
Matrix-vector system form
- 02
Eigenvalue and eigenvector calculation
- 03
Modal general solutions
- 04
Initial-condition constants
- 05
Stability and phase portraits
Worked example · free
Resolve two diagonal modes
- 1Identify both diagonal eigenpairs.
- 1Form the two modal terms.
- 1Use the initial state to determine constants.
- 1Interpret the mixed stability.
Key terms
- State vector
- A column collecting the dependent variables of a coupled system.
- Eigenvalue
- A scalar time-rate associated with one eigenvector direction.
- Eigenvector
- A nonzero direction preserved up to scaling by the coefficient matrix.
- Phase portrait
- A state-space plot showing trajectories and the system vector field.
- Asymptotic stability
- The property that nearby trajectories approach the equilibrium as time increases.
Linear ODE Systems and Eigenvalues FAQ
How do eigenpairs enter the solution?
Each eigenvector gives a state-space direction and its paired eigenvalue gives the exponential time factor. Independent modal terms are combined, then initial data determine their coefficients.
What makes an equilibrium unstable?
For the taught linear constant-coefficient systems, any eigenvalue with positive real part creates a growing mode. Typical initial states contain some of that mode, so trajectories move away from the equilibrium.
Are repeated eigenvalues always defective?
No. A repeated eigenvalue may still have enough linearly independent eigenvectors. Count its independent eigenvectors before deciding whether a generalised-vector construction is required in practice.
How can an eigenpair be verified efficiently?
Multiply the matrix by the proposed eigenvector and compare every component with the eigenvalue times that vector. This direct test catches an incorrect null-space vector or a vector paired with the wrong eigenvalue. Perform it before the pair is used in a modal solution or phase-portrait interpretation.
Exam move
Translate component equations into matrix form without swapping state order. Compute eigenvalues, solve separately for each eigenvector and verify the pairs. Build the general solution with column vectors, then solve the initial-condition system. State stability from all real parts and oscillation from imaginary parts. Use phase portraits to confirm qualitative direction, remembering that neither axis is time.
Begin each problem by writing the state vector and checking that every component equation matches the corresponding matrix row. Compute the characteristic polynomial with visible signs, then solve one null-space problem per eigenvalue and keep every vector paired with its own value. Test each pair immediately through matrix multiplication before it enters the general solution.
Write modes as column vectors multiplied by exponential factors and solve for constants by matching the complete initial vector, not one component at a time in isolation. For qualitative practice, classify sets of eigenvalues without calculating constants: mark their real parts, note any imaginary parts and predict node, saddle or spiral behaviour.
On a phase portrait, draw the eigendirections first and decide whether arrows move toward or away from the origin as time increases. Remember that a trajectory curve does not use time as either axis. Compare the portrait with a few evaluated state vectors from the formula. Treat repeated eigenvalues as a prompt to count independent eigenvectors, not as automatic evidence of a defective matrix.
Finish by differentiating the modal expression and matching matrix times state. A concise final interpretation should name stability, oscillation if present and which mode dominates long-run behaviour.
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