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Linear Algebra, Optimisation and Numerical Methods / Eigenvalues, Diagonalisation, and Dynamical Applications

Applications: Difference Equations, Markov Chains, and Fibonacci

This note is the first payoff of Stage 3's machinery: any discrete-time linear recursion uk+1=Auk\mathbf{u}_{k+1}=A\mathbf{u}_k — a portfolio evolving period by period, a Markov chain migrating between states, even the Fibonacci sequence — is solved exactly, in closed form, by diagonalisation. The eigenvalues determine everything about the long-run behaviour: which components grow, which decay, and — for a special and financially central case — which represent a genuine steady state.
Learning objectives. By the end of this note you should be able to:
  • solve uk+1=Auk\mathbf{u}_{k+1}=A\mathbf{u}_k in closed form using the eigendecomposition of AA;
  • identify a Markov matrix and explain why it always has λ=1\lambda=1 as an eigenvalue;
  • find the stationary distribution of a Markov chain as the λ=1\lambda=1 eigenvector;
  • solve the Fibonacci recursion via a 2×22\times2 matrix eigenvalue problem.

In this lesson

  1. Solving uk+1=Auk\mathbf{u}_{k+1}=A\mathbf{u}_k
  2. Markov chains and stationary distributions
  3. The Fibonacci sequence, via eigenvalues

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