The rank reduction procedure of Egerváry

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

[1]  D. Carlson,et al.  Matrix Decompositions Involving the Schur Complement , 1975 .

[2]  E. Spedicato,et al.  Abs Projection Algorithms: Mathematical Techniques for Linear and Nonlinear Equations , 1989 .

[3]  Aurél Galántai.  Rank reduction and bordered inversion , 2001 .

[4]  H. E. Goheen.  On a Lemma of Stieltjes on Matrices , 1949 .

[5]  Willem Hundsdorfer,et al.  A Second-Order Rosenbrock Method Applied to Photochemical Dispersion Problems , 1999, SIAM J. Sci. Comput..

[6]  Gene H. Golub,et al.  A Rank-One Reduction Formula and Its Applications to Matrix Factorizations , 1995, SIAM Rev..

[7]  E. Egerváry.  On rank-diminishing operations and their applications to the solution of linear equations , 1960 .

[8]  R. E. Cline,et al.  The Rank of a Difference of Matrices and Associated Generalized Inverses , 1976 .

[9]  Magyar Tudományos Akadémia. Nyelvtudományi Intézet,et al.  A Magyar Tudományos Akadémia Matematikai Kutató Intézetének közleményei = Труды Математического института Академии наук Венгрии = Publications of the Mathematical Institute of the Hungarian Academy of Sciences , 1956 .

[10]  T. N. E. Greville,et al.  Solutions of the Matrix Equation $XAX = X$, and Relations between Oblique and Orthogonal Projectors , 1974 .

[11]  W. J. Duncan,et al.  Elementary matrices and some applications to dynamics and differential equations , 1939 .

[12]  W. J. Duncan,et al.  Elementary matrices and some applications to dynamics and differential equations , 1939 .

[13]  P. Stanimirović.  Self-correcting iterative methods for computing ${2}$-inverses , 2003 .

[14]  Frazer.  Elementary Matrices: Frontmatter , 1938 .

[15]  T. Markham,et al.  A Generalization of the Schur Complement by Means of the Moore–Penrose Inverse , 1974 .

[16]  D. S. Tracy,et al.  Generalized Inverse Matrices: With Applications to Statistics , 1971 .

[17]  On some matrix equalities for generalized inverses with applications , 2009 .

[18]  Diane Valérie Ouellette.  Schur complements and statistics , 1981 .

[19]  Aurél Galántai,et al.  Rank reduction, factorization and conjugation , 2001 .

[20]  G. Styan,et al.  Equalities and Inequalities for Ranks of Matrices , 1974 .

[21]  J. B. Rosen,et al.  Lower Dimensional Representation of Text Data Based on Centroids and Least Squares , 2003 .

[22]  L. Guttman.  General theory and methods for matric factoring , 1944 .

[23]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[24]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[25]  E. Egerváry.  Über eine konstruktive Methode zur Reduktion einer Matrix auf die Jordansche Normalform , 1959 .

[26]  G. Stewart.  Conjugate direction methods for solving systems of linear equations , 1973 .

[27]  Patrick L. Odell,et al.  Matrix Theory: From Generalized Inverses to Jordan Form , 2007 .

[29]  Y. Takane,et al.  On the Wedderburn–Guttman theorem , 2005 .

[30]  P. L. Odell,et al.  Full Rank Factorization of Matrices , 1999 .

[31]  K. S. Banerjee.  Generalized Inverse of Matrices and Its Applications , 1973 .

[32]  C. G. Broyden.  On the numerical stability of Huang's and related methods , 1985 .

[33]  E. Egerváry.  Über eine Methode zur numerischen Lösung der Poissonschen Differenzengleichung für beliebige Gebiete , 1960 .

[34]  On the necessary and sufficient condition for the extended Wedderburn–Guttman theorem , 2009 .

[35]  Perturbation bounds for triangular and full rank factorizations , 2005 .

[36]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[37]  N. Urquhart.  Generalized Inverse Matrices (with Applications to Statistics) , 1973 .

[38]  A note on the generalized rank reduction , 2007 .

[39]  D. Carlson.  What are Schur complements, anyway? , 1986 .

[40]  H. W. Turnbull,et al.  Lectures on Matrices , 1934 .

[41]  A. Galántai.  Perturbations of Triangular Matrix Factorizations , 2003 .

[42]  A. Galántai.  Projectors and Projection Methods , 2003 .

[43]  Louis Guttman,et al.  Enlargement Methods for Computing the Inverse Matrix , 1946 .

[44]  Gene H. Golub,et al.  Rank Modifications of Semidefinite Matrices Associated with a Secant Update Formula , 1998, SIAM J. Matrix Anal. Appl..

[45]  Michael A. Saunders,et al.  Inertia-Controlling Methods for General Quadratic Programming , 1991, SIAM Rev..

[46]  A. Forsgren.  Inertia-controlling factorizations for optimization algorithms , 2002 .

[47]  Willem J. Heiser,et al.  Two Purposes for Matrix Factorization: A Historical Appraisal , 2000, SIAM Rev..

[48]  R. Cottle.  Manifestations of the Schur complement , 1974 .

[49]  E. Egerváry.  Über die Faktorisation von Matrizen und ihre Anwendung auf die Lösung von linearen Gleichungssystemen , 1955 .

[50]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[51]  The Reverse Bordering Method , 1994 .

[52]  E. Spedicato,et al.  A class of direct methods for linear systems , 1984 .

[53]  L. Guttman.  A necessary and sufficient formula for matric factoring , 1957 .

[54]  P. Rózsa,et al.  On eigenvectors and adjoints of modified matrices , 1981 .