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

Our work is motivated by a theorem proved by von Neumann: Let S1 and S2 be subspaces of a closed Hilbert space X and let x X. Then

limk(PS2PS1)k(x) = P S1S2(x),

where PS denotes the orthogonal projection of x onto the subspace S. We look at the linear algebra realization of the von Neumann theorem in n. The matrix A that represents the composition PS2PS1 has a form simple enough that the calculation of limkAkx 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.

orthogonal projections, von Neumann, best approximations
Mathematical Subject Classification 2010
Primary: 41A65
Secondary: 47N10
Received: 31 May 2012
Accepted: 2 June 2013
Published: 1 September 2013

Communicated by Jim Haglund
Rudy Joly
Johns Hopkins University
29 Hillview Avenue
Boston, MA 02131
United States
Marco López
Department of Mathematics
University of North Texas
1716 W. Hickory St. Apt 2
Denton, TX 76201
United States
Douglas Mupasiri
Department of Mathematics
University of Northern Iowa
220 Wright Hall
Cedar Falls, IA 50614-0506
United States
Michael Newsome
Jackson State University
8845 Hwy 12 West
Sallis, MS 39160
United States