power method

Repeatedly multiplying a vector by A and rescaling it, so it swings around to point along A's own dominant eigen-direction.

The power method produces the sequence $\mathbf x_{k+1}=A\mathbf x_k/\lVert A\mathbf x_k\rVert$ from a starting vector $\mathbf x_0$ with a nonzero component along the dominant eigenvector. Normalizing at every step keeps the sequence's LENGTH fixed while its DIRECTION converges to a unit eigenvector for $A$ 's dominant eigenvalue $\lambda_1$ , at a rate governed by $\lvert\lambda_2/\lambda_1\rvert$ .

Examples

For $A=\begin{pmatrix}4&1\\2&3\end{pmatrix}$ starting at $\mathbf x_0=(0,1)$ , a few power-iteration steps already swing the direction close to the dominant eigenvector for $\lambda_1=5$ .