Spectral characterization for von Neumann's iterative algorithm in $\mathbb{R}^n$

Abstract

Our work is motivated by a theorem proved by von Neumann: Let $S_1$ and $S_2$ be subspaces of a closed Hilbert space $X$ and let $x\in X$. Then

$$\underset{k\to \infty }{lim}{\left(P_{S_2}{P}_{S_1}\right)}^k\left(x\right)=P_{S_1\cap S_2}\left(x\right),$$

where $P_S$ denotes the orthogonal projection of $x$ onto the subspace $S$. We look at the linear algebra realization of the von Neumann theorem in ${ℝ}^n$. The matrix $A$ that represents the composition $P_{S_2}{P}_{S_1}$ has a form simple enough that the calculation of ${lim}_{k\to \infty }{A}^{k}x$ becomes easy. However, a more interesting result lies in the analysis of eigenvalues and eigenvectors of $A$ and their geometrical interpretation. A characterization of such eigenvalues and eigenvectors is shown for subspaces with dimension $n-1$.