CS223 · Annotated SlidesChapter 5 — Complex Eigenvalues and Dynamical Systems

Chapter 5 · Annotated slides

Complex Eigenvalues and Dynamical Systems

Complex eigenvalues and eigenvectors in conjugate pairs, rotation–scaling matrices, long-term behaviour of xₖ₊₁ = Axₖ, population models and steady-state vectors.

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For real A, complex eigenvalues come in conjugate pairs λ, λ̄ with eigenvectors x, x̄.

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Slide 4: Long-term evaluation: Example

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Long-term behaviour of xₖ = Aᵏx₀: write x₀ in the eigenvector basis; terms with |λ| > 1 grow, |λ| < 1 decay.

Slide 5: eigenfunctions of the differential equation

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Slide 6: Example:

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Slide 13: Example:

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Steady-state vector of a stochastic (Markov) matrix P: Pq = q with entries summing to 1: solve (P − I)q = 0.

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How do you find the steady-state vector of a stochastic matrix P?

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Slide 17: k

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Slide 18: Example:

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