Rank decomposition in zero pattern matrix algebras

Czechoslovak Mathematical Journal

Harm Bart¹  Bernd Silbermann²  Torsten Ehrhardt³ 

[1] Econometric Institute, Erasmus University Rotterdam, Burgemeester Oudlaan 50, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands;Fakultät für Mathematik, Technische Universität Chemnitz, Reichenhainer Str. 39, 09107 Chemnitz, Germany;Mathematics Department, University of California, 1156 High St., Santa Cruz, CA-95064, California, USA

关键词: block upper triangularity;    additive decomposition;    rank constraints;    zero pattern matrix algebra;    preorder;    partial order;    Hasse diagram;    rooted tree;    out-tree;    in-tree;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see H. Bart, A. P. M. Wagelmans (2000). The proof involves elements from integer programming and employs Farkas' lemma. The algebra of block upper triangular matrices can be viewed as a matrix algebra determined by a pattern of zeros. The present note is concerned with the question whether the decomposition result referred to above can be extended to other zero pattern matrix algebras. It is shown that such a generalization does indeed hold for certain digraphs determining the pattern of zeros. The digraphs in question can be characterized in terms of forests, i.e., disjoint unions of rooted trees.

【 授权许可】

Unknown