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Linear Algebra, Optimisation and Numerical Methods / Eigenvalue Computation Algorithms

Power Iteration and the Rayleigh Quotient

LA-I's characteristic-polynomial method solves det⁡(A−λI)=0\det(A-\lambda I)=0 directly — practical for 2×22\times2 and 3×33\times3 matrices, but for larger nn, this polynomial has no closed-form root formula in general. This note opens Stage 4's eigenvalue-computation toolkit with the simplest iterative alternative: power iteration, which finds the dominant eigenvalue by repeatedly applying AA to a vector and watching it align with the dominant eigen-direction — precisely the mechanism LA-I's dynamical-systems applications already exploited, now used deliberately as a computational tool.
Learning objectives. By the end of this note you should be able to:
  • run power iteration by hand on a small matrix;
  • explain the algorithm's convergence rate in terms of the eigenvalue gap;
  • compute the Rayleigh quotient and explain why it improves the eigenvalue estimate;
  • identify when power iteration converges slowly or fails.

In this lesson

  1. The algorithm
  2. The Rayleigh quotient

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