Single-vector iteration, also known as the power method, is one of the simplest ways to approximate the dominant eigenvalue and eigenvector of a matrix.

For a matrix \(A\), the basic idea is:

  • \(A \in \mathbb{R}^{n \times n}\), with \(n\) distinct real eigenvalues
\[|\lambda_1| > |\lambda_2| > \dots > |\lambda_n|\]
  • And the corresponding eigenvectors are:
\[e_1, e_2, \dots, e_n\]
  • We can express a random starting vector \(x_0\) as a linear combination of eigenvectors:
\[x_0 = c_1 e_1 + c_2 e_2 + \dots + c_n e_n = \sum_{i=1}^{n} c_i e_i\]
  • By definition of an eigenpair:
\[A e_i = \lambda_i e_i, \quad i=1,\dots,n\]
  • Repeated multiplication by \(A\) gives:
\[x_1 = A x_0 = A \sum_{i=1}^{n}c_ie_i =\sum_{i=1}^{n}c_i A e_i = \sum_{i=1}^{n}c_i \lambda_i e_i\] \[x_p = A^p x_0 = \sum_{i=1}^{n}c_i \lambda_i^p e_i\]
  • After repeated multiplication, the dominant component remains:
\[x_p = \lambda_1^p \left[c_1 e_1 + \sum_{i=2}^{n} c_i \left(\frac{\lambda_i}{\lambda_1}\right)^p e_i \right] \approx \lambda_1^p c_1 e_1\] \[x_{p+1} \approx \lambda_1^{p+1} c_1 e_1 = \lambda_1 x_p\]
  • Since $x_{p+1}$ is only approximately a scalar multiple of $x_p$, we estimate the scalar by projecting one vector onto the other:
\[\lambda_1 \approx \frac{x_p^T x_{p+1}}{x_p^T x_p}\] \[e_1 \approx x_p\]
  • To prevent underflow or overflow, we normalize the eigenvector estimate:
\[x_p = \frac{x_p}{(x_p^Tx_p)^{1/2}}\]

The explanation above was taken from Power method of eigenvalues - Lesson 30 - Numerical Methods for Engineers.

The animation below illustrates the iteration process.