How do you find the steady-state vector of a stochastic matrix P?
Compare with the model answer
Solve (P − I)x = 0, pick a nonzero solution, and scale it so its entries add up to 1.
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.
How to read this page
Every original slide is shown unchanged. Yellow marks on a slide point to a numbered note: hover or tap a mark to jump to its note, and hover a note to see its marks. Questions tagged Past exam appeared on a real paper; Prediction questions are likely angles, not confirmed. Pick an answer to each question: a wrong pick tells you why it is wrong, so you can try again.
Slide 1
Slide 2
Notes
For real A, complex eigenvalues come in conjugate pairs λ, λ̄ with eigenvectors x, x̄.
Slide 3
Long-term evaluation: Example
Notes
Long-term behaviour of xₖ = Aᵏx₀: write x₀ in the eigenvector basis; terms with |λ| > 1 grow, |λ| < 1 decay.
eigenfunctions of the differential equation
Example:
Slide 7
Slide 8
Slide 9
Slide 10
Slide 11
Slide 12
Example:
Slide 14
Slide 15
Notes
Steady-state vector of a stochastic (Markov) matrix P: Pq = q with entries summing to 1: solve (P − I)q = 0.
Exam angle
Slide 16
k
Example:
