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Real and integer Wedderburn rank reduction formulas for matrix decompositions

Mahdavi Amiri, N ; Sharif University of Technology | 2015

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  1. Type of Document: Article
  2. DOI: 10.1080/10556788.2014.1002192
  3. Publisher: Taylor and Francis Ltd , 2015
  4. Abstract:
  5. 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 ZTZ and the WTW 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
  6. Keywords:
  7. Banded factorization ; Biconjugation process ; Hessenberg factorization ; Linear Diophantine system ; Quadratic Diophantine equation ; Smith norma l form ; Wedderburn rank reduction formula ; Factorization ; Linear algebra ; Linear transformations ; Numerical methods ; Biconjugation ; Diophantine equation ; Diophantine system ; Matrix decomposition ; Wedderburn rank reductions ; WZ factorization ; Matrix algebra
  8. Source: Optimization Methods and Software ; Volume 30, Issue 4 , 2015 , Pages 864-879 ; 10556788 (ISSN)
  9. URL: http://www.tandfonline.com/doi/abs/10.1080/10556788.2014.1002192?journalCode=goms20