Power Iteration for Dominant Eigenvalue

Math Linear Algebra • Linear Transformations and Eigenvalues

Written by STEM Calculators Team

Approximate the dominant eigenvalue (largest \(|\lambda|\)) and a corresponding eigenvector using power iteration: \(\mathbf{v}_{k+1}=\dfrac{A\mathbf{v}_k}{\lVert A\mathbf{v}_k\rVert}\), with eigenvalue estimate \(\lambda_k=\mathbf{v}_k^{\mathsf{T}}A\mathbf{v}_k\) (Rayleigh quotient for normalized \(\mathbf{v}_k\)). Includes convergence checks, an iteration table, and a convergence plot.

Matrix size

Iterations

Convergence tol

Display decimals

Steps

Use shift \(\mu\) (runs power iteration on \(A-\mu I\))

\(\mu\)

Show 2D vector visual (only for 2×2)

Matrix \(A\)

A is 2×2

Inputs accept -3.5, 2e-4, fractions 7/3, and constants pi, e.

Initial vector \(\mathbf{v}_0\)

Dimension matches the matrix size.

If \(\mathbf{v}_0\) has no component in the dominant eigenvector direction, the method may stall or converge to a different mode.

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Results

Dominant eigenvalue estimate

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Eigenvector estimate

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Residual \(\lVert A\mathbf{v}-\lambda\mathbf{v}\rVert\)

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Convergence

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Iteration summary

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Iteration table

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Step-by-step

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Theory

Power iteration idea

Power iteration is a simple method for approximating the eigenvalue of largest magnitude (the dominant eigenvalue) and a corresponding eigenvector. Assume \(A\in\mathbb{R}^{n\times n}\) has an eigenvalue \(\lambda_1\) with \(|\lambda_1| > |\lambda_2|\ge \cdots \ge |\lambda_n|\), and let \(\mathbf{q}_1\) be an eigenvector for \(\lambda_1\).

If you write a starting vector \(\mathbf{v}_0\) as a combination of eigenvectors \(\mathbf{v}_0 = c_1\mathbf{q}_1 + c_2\mathbf{q}_2 + \cdots\), then repeated multiplication gives \[ A^k\mathbf{v}_0 = c_1\lambda_1^k\mathbf{q}_1 + c_2\lambda_2^k\mathbf{q}_2 + \cdots. \] Because \(|\lambda_1|^k\) grows fastest, the direction of \(A^k\mathbf{v}_0\) tends to align with \(\mathbf{q}_1\) as long as \(c_1\neq 0\) (meaning \(\mathbf{v}_0\) has some component in the dominant eigenvector direction).

Normalization is essential: \(\mathbf{v}_{k+1} = \dfrac{A\mathbf{v}_k}{\lVert A\mathbf{v}_k\rVert}\) keeps vectors from blowing up or underflowing.

Eigenvalue estimates and convergence checks

After computing a normalized iterate \(\mathbf{v}_k\), a common eigenvalue estimate is the Rayleigh quotient \[ \lambda_k = \frac{\mathbf{v}_k^{\mathsf T}A\mathbf{v}_k}{\mathbf{v}_k^{\mathsf T}\mathbf{v}_k}. \] If \(\mathbf{v}_k\) is normalized so that \(\mathbf{v}_k^{\mathsf T}\mathbf{v}_k = 1\), this simplifies to \(\lambda_k = \mathbf{v}_k^{\mathsf T}A\mathbf{v}_k\).

A useful way to measure how close \((\lambda_k,\mathbf{v}_k)\) is to a true eigenpair is the residual: \[ r_k = \lVert A\mathbf{v}_k - \lambda_k\mathbf{v}_k\rVert. \] When \(r_k\) is small, \(\mathbf{v}_k\) behaves like an eigenvector and \(\lambda_k\) behaves like its eigenvalue. Many implementations stop when \(r_k\) falls below a tolerance, or when successive eigenvalue estimates change very little: \[ |\lambda_k - \lambda_{k-1}| \le \text{tol}. \]

Slow convergence happens when \(|\lambda_2/\lambda_1|\) is close to 1. The closer that ratio is to 1, the more iterations are needed.

Shifts and finding other eigenvalues

Plain power iteration targets the eigenvalue of largest magnitude. To aim at other parts of the spectrum, more advanced variants are used. A basic “shift” idea modifies the matrix to \(B=A-\mu I\). Power iteration on \(B\) finds the eigenvalue of \(B\) with largest magnitude, which corresponds to the eigenvalue of \(A\) farthest from \(\mu\). This can be helpful in some cases, but it is not the same as “closest to \(\mu\)”.

In university-level numerical linear algebra, the most common approach for targeting a specific eigenvalue is inverse iteration or shift-invert: apply power iteration to \((A-\mu I)^{-1}\) (when it exists). That method tends to converge to the eigenvalue nearest \(\mu\).

Real-world tease: PageRank can be viewed as finding a dominant eigenvector of a stochastic matrix.

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