The rank reduction procedure of Egerváry

Galántai, Aurél

doi:10.1007/s10100-009-0124-0

Cited by 8 publications

(3 citation statements)

References 41 publications

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“…The Wedderburn-Egerváry rank reduction formula characterizes the rank-one modifications that reduce the rank of a given rectangular matrix X ∈ R n×p exactly by one, see [7]. An account of the history of this theory is given in [11]. The precise statement is as follows:…”

Section: Relation To Existing Workmentioning

confidence: 99%

A geometric note on subspace updates and orthogonal matrix decompositions under rank-one modifications

Zimmermann

2017

Preprint

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In this work, we consider rank-one adaptations Xnew = X + ab T of a given matrix X ∈ R n×p with known matrix factorization X = U W , where U ∈ R n×p is column-orthogonal, i.e. U T U = I. Arguably the most important methods that produce such factorizations are the singular value decomposition (SVD), where X = U W = U ΣV T , and the QR-decomposition, where X = U W = QR. An elementary approach to produce a column-orthogonal matrix Unew, whose columns span the same subspace as the columns of the rank-one modified Xnew = X + ab T is via applying a suitable coordinate change such that in the new coordinates, the update affects a single column and subsequently performing a Gram-Schmidt step for reorthogonalization. This may be interpreted as a rank-one adaptation of the U -factor in the SVD or a rank-one adaptation of the Q-factor in the QR-decomposition, respectively, and leads to a decomposition for the adapted matrix Xnew = UnewWnew. By using a geometric approach, we show that this operation is equivalent to traveling from the subspace S = ran(X) to the subspace Snew = ran(Xnew) on a geodesic line on the Grassmann manifold and we derive a closed-form expression for this geodesic. In addition, this allows us to determine the subspace distance between the subspaces S and Snew without additional computational effort. Both Unew and Wnew are obtained via elementary rank-one matrix updates in O(np) time for n p. Possible fields of applications include subspace estimation in computer vision, signal processing, update problems in data science and adaptive model reduction.

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Section: Relation To Existing Workmentioning

confidence: 99%

A geometric note on subspace updates and orthogonal matrix decompositions under rank-one modifications

Zimmermann

2017

Preprint

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“…The sufficient part of this result is included in Wedderburn's book and was later extended to block matrices by Guttman [14,15]. Galántai [2,[9][10][11] gave several results concerning the rank reduction algorithm developed by Egerváry. Wedderburn showed that subtracting rank one matrices of the form ω −1 Axy T A from a matrix A resulted in a matrix with rank one less than that of A, if ω = y T Ax = 0 [16,22]. The converse is also true [16].…”

Section: Introductionmentioning

confidence: 96%

Real and integer Wedderburn rank reduction formulas for matrix decompositions

Mahdavi–Amiri

Golpar-Raboky

2015

Optimization Methods and Software

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The Wedderburn rank reduction formula is a powerful method for developing matrix factorizations and many fundamental numerical linear algebra processes. We present a new interpretation of the Wedderburn rank reduction formula and its associated biconjugation process and see a more extensive result of the formula. In doing this, we propose a new formulation based on the null space transformations on rows and columns of A simultaneously, and show several matrix factorizations that can be derived from the Wedderburn rank reduction formula. We also present a generalization of the biconjugation process and compute banded and Hessenberg factorizations. Using the new formulation, we compute the WZ and ZW factorizations of a nonsingular matrix as well as the Z T Z and the W T W factorizations of a symmetric positive definite matrix. Then, we develop the integer Wedderburn rank reduction formula and its integer biconjugation process and present a class of algorithms for computing all integer matrix factorizations such as the integer LU factorization and the Smith normal form of an arbitrary integer matrix.

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“…The paper of Galántai (2010) gives a survey of the recent results on Egerváry's rank reduction algorithm and reports some applications in optimization algorithms. Hegedűs (2010) provides an interesting review on the Purcell method that was significantly improved by Egerváry in the context of his rank reduction procedure.…”

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confidence: 99%

Special issue in honour of Jenő Egerváry

Csendes

Galántai

Szeidl

2009

Cent Eur J Oper Res

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) was a prominent scientist of pure and applied mathematics. He is best known for his contribution to the Hungarian method. However, his research activity included much more than that and he obtained many important and genuine results in other areas of mathematics and its applications. He was also an outstanding teacher and a key person of Hungarian applied mathematics until 1958, when he committed suicide. This was the aftermath of the 1956 revolution of Hungary with a harsh oppression. Not long after Egerváry's death, his research department was dissolved and by the early seventies he and his results were rarely mentioned in mathematical circles apart from his relation to the Hungarian method. At the end of the eighties his linear algebra related research came into light due to its connection T. Csendes (B)

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The rank reduction procedure of Egerváry

Cited by 8 publications

References 41 publications

A geometric note on subspace updates and orthogonal matrix decompositions under rank-one modifications

A geometric note on subspace updates and orthogonal matrix decompositions under rank-one modifications

Real and integer Wedderburn rank reduction formulas for matrix decompositions

Special issue in honour of Jenő Egerváry

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