A Continuous Selection of the Metric Projection in Matrix Spaces

Berens, Hubert; Finzel, M.

doi:10.1007/978-3-0348-6656-9_2

Cited by 6 publications

(10 citation statements)

References 2 publications

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“…j .A//. The orthogonal matrices U A and V A from the SVD are not unique in the general case (for a relation between orthogonal matrices from two singular value decompositions of A, see Lemma in [19]; compare Problem B.11 in [20, p. 334] and [21]). Let A D P † A Q T be another SVD of A, let † A D diag.…”

Section: Auxiliary Known Results and Notationmentioning

confidence: 99%

On a sub‐Stiefel Procrustes problem arising in computer vision

Cardoso

Ziętak

2015

Numerical Linear Algebra App

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A sub-Stiefel matrix is a matrix that results from deleting simultaneously the last row and the last column of an orthogonal matrix. In this paper, we consider a Procrustes problem on the set of sub-Stiefel matrices of order n. For n D 2, this problem has arisen in computer vision to solve the surface unfolding problem considered by R. Fereirra, J. Xavier and J. Costeira. An iterative algorithm for computing the solution of the sub-Stiefel Procrustes problem for an arbitrary n is proposed, and some numerical experiments are carried out to illustrate its performance. For these purposes, we investigate the properties of sub-Stiefel matrices. In particular, we derive two necessary and sufficient conditions for a matrix to be sub-Stiefel. We also relate the sub-Stiefel Procrustes problem with the Stiefel Procrustes problem and compare it with the orthogonal Procrustes problem.

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Section: Auxiliary Known Results and Notationmentioning

confidence: 99%

On a sub‐Stiefel Procrustes problem arising in computer vision

Cardoso

Ziętak

2015

Numerical Linear Algebra App

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“…diag ( T , H,,-'), it is w_ell-Jnown that the unitary matrices U and vestablish just another SVD of B ; i.e., B = UCV, cf. e.g., [2]. It remains to show that ACj = zjCj, 1 I j I r. Indeed, for 1 5 j 2 r…”

Section: Lemma Ij'b Belongs To Ma There Existsmentioning

confidence: 98%

A Problem in Linear Matrix Approximation

Berens

Finzel

1995

Mathematische Nachrichten

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Let A be a normal operator in g ( H ) , H a complex Hilbert space, and let g A = 2 { A X -X A : X E g ( H ) } be the commutator subspace of B ( H ) associated with A . If B in g(H)commutes with A, then B is orthogonal to 9, with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in 9,. This was proved by J. ANDERSON in 1973 and extended by P. J. MAHER with respect to the Schatten p-norm recently.We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in W , and prove that the metric projection of H onto . %!A is continuous.

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“…Approximation in matrix spaces proved to be rich in providing simple, but impressive examples as the papers [1,4,5,7] show. We want to take up the results in [3,4], extend, and re-prove them.…”

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confidence: 98%

“…For a proof see [3]. Note that we used the subscript to indicate that we consider the spectral norm on C n_n .…”

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confidence: 99%