Rank  reduction and bordered inversion

Galántai, Aurél

doi:10.18514/mmn.2001.42

Cited by 4 publications

(1 citation statement)

References 11 publications

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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: 99%

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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