#### Vol. 6, No. 2, 2013

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

### Rudy Joly, Marco López, Douglas Mupasiri and Michael Newsome

Vol. 6 (2013), No. 2, 243–249
##### 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 $n-1$.

##### Keywords
orthogonal projections, von Neumann, best approximations
Primary: 41A65
Secondary: 47N10