 Power iteration

Power iteration finds a dominant eigenvalue and its eigenvector by repeatedly applying a matrix and rescaling the resulting vector. The essential idea is selective amplification: components along different eigenvectors grow at different rates, so one direction can eventually dominate. It is the computational form of the power method and is useful when applying a large sparse matrix to a vector is cheap but computing its whole spectral decomposition is expensive.

For a square matrix over the real numbers or complex numbers $A$, choose a unit vector $x_0$ and repeatHere $\Vert \cdot {\Vert }_2$ is the Euclidean norm. If $y_{k+1}=0$, stop before division: the current vector is an eigenvector for the zero eigenvalue, and this iteration cannot continue. Otherwise, in exact arithmetic,Repeated normalization keeps the stored vectors at manageable size without changing their directions; it does not guarantee that an individual multiplication cannot exceed the range of floating-point arithmetic. The dominant eigenvalue means largest modulus, which need not mean largest signed real value.^[ref]

Power iteration is a numerical method used to find the dominant eigenvalue and its corresponding eigenvector of a matrix. This technique is particularly effective for large, sparse matrices, where traditional methods like direct diagonalization may be computationally expensive or impractical. ### How Power Iteration Works: 1. **Initialization**: Start with a random vector \( \mathbf{b_0} \) (which should not be orthogonal to the eigenvector corresponding to the dominant eigenvalue).