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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
∈
X .
Then
lim k → ∞ ( P S 2 P S 1 ) k ( x )
= P
S 1 ∩ S 2 ( x ) ,
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 → ∞ 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
Mathematical Subject Classification 2010
Primary: 41A65
Secondary: 47N10
Milestones
Received: 31 May 2012
Accepted: 2 June 2013
Published: 1 September 2013
Communicated by Jim Haglund