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Home/Math Visualization/Power Iteration Eigenvalue Convergence

Power Iteration Eigenvalue Convergence

Power iteration repeatedly multiplies a vector by A and normalizes it, revealing the eigenvector associated with the dominant eigenvalue when |lambda1|>|lambda2|. This simulator uses a 2D symmetric matrix with visible eigenvectors, so each normalized iterate can be plotted on the unit circle. The Rayleigh quotient estimates the eigenvalue, the eigen residual ||Ax-rho x|| measures how close the current vector is to an eigenvector, and the spectral gap ratio |lambda2/lambda1| explains why convergence can be fast or painfully slow.

Who it's for: Numerical linear algebra, scientific computing, data science, graph algorithms, and applied mathematics courses.

Key terms

  • Power iteration
  • Dominant eigenvector
  • Rayleigh quotient
  • Spectral gap
  • Eigen residual
  • Convergence rate

The demo uses a 2D symmetric matrix so the true eigenvectors are visible. When the spectral ratio is close to one, convergence slows; if the initial vector is nearly orthogonal to the dominant eigenvector, the method starts with very little useful component.

Live graphs

Matrix eigen-structure

6
2.2
28°

Power iteration

112°
8

Measured values

Rayleigh quotient5.99996
Spectral ratio |λ2/λ1|0.3667
Eigen residual0.01181
Angle error0.1781°

How it works

Power iteration visualizer for dominant eigenvector convergence, spectral gap, Rayleigh quotient, and eigen residual.

Key equations

x_{k+1}=A x_k / ||A x_k||, ρ_k=(x_k^T A x_k)/(x_k^T x_k)
For |λ1|>|λ2|, direction error decays roughly like |λ2/λ1|^k

Frequently asked questions

What controls the convergence rate of power iteration?
For a diagonalizable matrix with a unique dominant eigenvalue, the direction error usually decays like |lambda2/lambda1|^k. A larger spectral gap means a smaller ratio and faster convergence.
Why use the Rayleigh quotient?
Once the vector is close to an eigenvector, rho=x^T A x / x^T x gives a natural eigenvalue estimate. The residual ||Ax-rho x|| then checks whether the pair is actually close to an eigenpair.