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.

Power Iteration Eigenvalue Convergence

Visualize dominant eigenvector convergence: spectral gap ratio, Rayleigh quotient, eigen residual, and normalized power iterates on the unit circle.

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

About this model

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

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.

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Power Iteration Eigenvalue Convergence

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About this model

How it works

Key equations

Frequently asked questions

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Frequently asked questions