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The rank reduction procedure of Egerváry

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Abstract

Here we give a survey of results concerning the rank reduction algorithm developed by Egerváry between 1953 and 1958 in a sequence of papers.

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References

  • Abaffy J, Broyden CG, Spedicato E (1984) A class of direct methods for linear systems. Numerische Mathematik 45: 361–376

    Article  Google Scholar 

  • Abaffy JE, Spedicato E (1989) ABS-projection algorithms: mathematical techniques for linear and nonlinear algebraic equations. Ellis Horwood, Chichester

    Google Scholar 

  • Ben-Israel A, Greville TNE (1974) Generalized inverses: theory and applications. Wiley, New York

    Google Scholar 

  • Brezinski C, Cecchi MM, Redivo-Zaglia M (1994) The reverse bordering method. SIAM J Matrix Anal Appl 15: 922–937

    Article  Google Scholar 

  • Broyden CG (1985) On the numerical stability of Huang’s and related methods. JOTA 47: 401–412

    Article  Google Scholar 

  • Broyden CG (1997) The Gram-Schmidt method—a hierarchy of algorithms. In: Sydow A (ed) 15th IMACS world congress on scientific computation, modelling and applied mathematics, vol 2. Numerical Mathematics, Wissenschaft and Technik, Berlin, pp 545–550

  • Bunch JR, Parlett BN (1971) Direct methods for solving symmetric indefinite systems of linear equations. SIAM J Numer Anal 8: 639–655

    Article  Google Scholar 

  • Carlson D (1975) Matrix decompositions involving the Schur complement. SIAM J Appl Math 28: 577–587

    Article  Google Scholar 

  • Carlson D (1986) What are Schur complements, anyway?. Linear Algebra Appl 74: 257–275

    Article  Google Scholar 

  • Carlson D, Haynsworth E, Markham T (1974) A generalization of the Schur complement by means of the Moore-Penrose inverse. SIAM J Appl Math 26: 169–179

    Article  Google Scholar 

  • Cline RE, Funderlic RE (1979) The rank of a difference of matrices and associated generalized inverses. Linear Algebra Appl 24: 185–215

    Article  Google Scholar 

  • Chu MT, Funderlic RE, Golub GH (1995) A rank-one reduction formula and its applications to matrix factorizations. SIAM Rev 37: 512–530

    Article  Google Scholar 

  • Chu MT, Funderlic RE, Golub GH (1998) Rank modifications of semidefinite matrices associated with a secant update formula. SIAM J Matrix Anal Appl 20: 428–436

    Google Scholar 

  • Cottle RW (1974) Manifestations of the Schur complement. Linear Algebra Appl 8: 189–211

    Article  Google Scholar 

  • Egerváry J (1953) Mátrixok diadikus elôállításán alapuló módszer bilineáris alakok transzformációjára és lineáris egyenletrendszerek megoldására (A method based on the dyadic representation of matrices for the transformation of bilinear forms and for solving systems of linear equations, in Hungarian). MTA Alkalmazott Matematikai Intézetének Közleményei 2: 11–32

    Google Scholar 

  • Egerváry J (1953b) On a property of the projector matrices and its application to the canonical representation of matrix functions. Acta Sci Math 15: 1–6

    Google Scholar 

  • Egerváry J (1954) On a lemma of Stieltjes on matrices. Acta Sci Math 15: 99–103

    Google Scholar 

  • Egerváry J (1955) Über die Faktorisation von Matrizen und ihre Anwendung auf die Lösung von linearen Gleichungssystemen. Z Angew Mathematik Mechanik 35: 111–118

    Article  Google Scholar 

  • Egerváry J (1956) Régi és új módszerek lineáris egyenletrendszerek megoldására (Old and new methods for solving linear equations, in Hungarian). A Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei (Publications of the Mathematical Institute of the Hungarian Academy of Sciences) 1: 109–123

    Google Scholar 

  • Egerváry J (1956b) Auflösung eines homogenen linearen diophantischen Gleichungssystems mit Hilfe von Projektormatrizen. Publicationes Math 4: 481–483

    Google Scholar 

  • Egerváry J (1956) Az inverz mátrix általánosítása (On a generalized inverse for matrices, in Hungarian). A Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei (Publications of the Mathematical Institute of the Hungarian Academy of Sciences) 1: 315–324

    Google Scholar 

  • Egerváry J (1957) Über eine Verallgemeinerung der Purcellschen Methode zur Auflösung linearer Gleichungssyteme. Österr Ingeniur-Archiv 11: 249–251

    Google Scholar 

  • Egerváry J (1959) Über einige konstruktive Methode zur Reduktion einer Matrix auf die Jordansche Normalform. Acta Math Acad Hung 10: 31–54

    Article  Google Scholar 

  • Egerváry E (1960a) On rank-diminishing operations and their applications to the solution of linear equations. ZAMP 11: 376–386

    Article  Google Scholar 

  • Egerváry J (1960b) Über eine Methode zur numerischen Lösung der Poissonschen Differenzengleichung für beliebige Gebiete. Acta Math Acad Sci Hung 11: 341–361

    Article  Google Scholar 

  • Elsner L, Rózsa P (1981) On eigenvectors and adjoints of modified matrices. Linear Multilinear Algebra 10: 235–247

    Article  Google Scholar 

  • Forsgren A (2002) Inertia-controlling factorizations for optimization algorithms. Appl Numer Math 43: 91–107

    Article  Google Scholar 

  • Frazer A, Duncan J, Collar R (1938) Elementary matrices and some applications to dynamics and differential equations. Cambridge University Press, Cambridge

    Google Scholar 

  • Galántai A (1997–1999) Rank reduction and conjugation. Acta Technica Acad Sci Hung 108(1–2): 107–130

    Google Scholar 

  • Galántai A (2000) Rank reduction and conjugation. Math Notes Miskolc 1: 11–33

    Google Scholar 

  • Galántai A (2001a) Rank reduction and bordered inversion. Math Notes Miskolc 2: 117–126

    Google Scholar 

  • Galántai A (2001b) Rank reduction, factorization and conjugation. Linear Multilinear Algebra 49: 195–207

    Article  Google Scholar 

  • Galántai A (2003a) Rank reduction: theory and applications. Int J Math, Game Theory Algebra 13(2): 173–189

    Google Scholar 

  • Galántai A (2003b) Perturbations of triangular matrix factorizations. Linear Multilinear Algebra 51: 175–198

    Article  Google Scholar 

  • Galántai A (2004) Projectors and projection methods. Kluwer, Dordrecht

    Google Scholar 

  • Galántai A (2005) Perturbation bounds for triangular and full rank factorizations. Comput Math Appl 50: 1061–1068

    Article  Google Scholar 

  • Galántai A (2007) A note on the generalized rank reduction. Acta Math Hungar 116(3): 239–246

    Article  Google Scholar 

  • Gill PE, Murray W, Saunders MA, Wright MH (1991) Inertia-controlling methods for general quadratic programming. SIAM Rev 33: 1–36

    Article  Google Scholar 

  • Greville TNE (1974) Solutions of the matrix equation XAXX, and relations between oblique and orthogonal projectors. SIAM J Appl Math 26: 828–832

    Article  Google Scholar 

  • Guttman L (1944) General theory and methods for matric factoring. Psychometrika 9: 1–16

    Article  Google Scholar 

  • Guttman L (1946) Enlargement methods for computing the inverse matrix. Ann Math Statist 17: 336–343

    Article  Google Scholar 

  • Guttman L (1957) A necessary and sufficient formula for matrix factoring. Psychometrika 22: 79–91

    Article  Google Scholar 

  • Hegedűs C (2009) Private communication

  • Householder AS (1964) The theory of matrices in numerical analysis. Blaisdell, New York

    Google Scholar 

  • Hubert L, Meulman J, Heiser W (2000) Two purposes for matrix factorization: A historical appraisal. SIAM Rev 42: 68–82

    Article  Google Scholar 

  • Marsaglia G, Styan GPH (1974) Equalities and inequalities for ranks of matrices. Linear and Multilinear Algebra 2: 269–292

    Article  Google Scholar 

  • Ouellette DV (1981) Schur complements and statistics. Linear Algebra Appl 36: 187–295

    Article  Google Scholar 

  • Park H, Jeon M, Rosen JB (2003) Lower dimensional representation of text data based on centroids and least squares. BIT Numer Math 43: 427–448

    Article  Google Scholar 

  • Piziak R, Odell PL (1999) Full rank factorization of matrices. Math Mag 72(3): 193–201

    Article  Google Scholar 

  • Piziak R, Odell PL (2007) Matrix theory: from generalized inverses to Jordan form. Chapman & Hall/CRC, Taylor & Francis Group, UK

    Google Scholar 

  • Pringle RM, Rayner AA (1971) Generalized inverse matrices with applications to statistics. Griffin, London

    Google Scholar 

  • Radić M (1966) Some contributions to the inversion of rectangular matrices. Glasnik Matematićki 21: 23–37

    Google Scholar 

  • Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York

    Google Scholar 

  • Saad Y (1996) Iterative methods for sparse linear systems. PWS Publishing Company, Boston

    Google Scholar 

  • Stanimirović PS (2003) Self-correcting iterative methods for computing {2}-inverses. Arch Math (Brno) 39: 27–36

    Google Scholar 

  • Stewart GW (1973) Conjugate direction methods for solving systems of linear equations. Numerische Math 21: 285–297

    Article  Google Scholar 

  • Takane Y, Yanai H (2005) On the Wedderburn–Guttman theorem. Linear Algebra Appl 410: 267–278

    Article  Google Scholar 

  • Takane Y, Yanai H (2007) Alternative characterizations of the extended Wedderburn–Guttman theorem. Linear Algebra Appl 422: 701–711

    Article  Google Scholar 

  • Takane Y, Yanai H (2009) On the necessary and sufficient condition for the extended Wedderburn–Guttman theorem. Linear Algebra Appl doi:10.1016/j.laa.2008.12.032

  • Tian Y, Styan GPH (2008) On some matrix equalities for generalized inverses with applications. Linear Algebra Appl doi:10.1016/j.laa.2008.12.005

  • Wedderburn JHM (1934) Lectures on matrices. Amer Math Soc Colloquium Publications, XVII

  • Zhang L (1998) Computing inertias of KKT matrix and reduced Hessian via the ABS algorithm, unpublished manuscript, Department of Applied Mathematics, Dalian University of Technology, Dalian, China, pp 1–14

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Correspondence to Aurél Galántai.

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Galántai, A. The rank reduction procedure of Egerváry. Cent Eur J Oper Res 18, 5–24 (2010). https://doi.org/10.1007/s10100-009-0124-0

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