# Rank decomposition in zero pattern matrix algebras

Harm Bart; Torsten Ehrhardt; Bernd Silbermann

Czechoslovak Mathematical Journal (2016)

- Volume: 66, Issue: 3, page 987-1005
- ISSN: 0011-4642

## Access Full Article

top## Abstract

top## How to cite

topBart, Harm, Ehrhardt, Torsten, and Silbermann, Bernd. "Rank decomposition in zero pattern matrix algebras." Czechoslovak Mathematical Journal 66.3 (2016): 987-1005. <http://eudml.org/doc/286807>.

@article{Bart2016,

abstract = {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.},

author = {Bart, Harm, Ehrhardt, Torsten, Silbermann, Bernd},

journal = {Czechoslovak Mathematical Journal},

keywords = {block upper triangularity; additive decomposition; rank constraints; zero pattern matrix algebra; preorder; partial order; Hasse diagram; rooted tree; out-tree; in-tree},

language = {eng},

number = {3},

pages = {987-1005},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Rank decomposition in zero pattern matrix algebras},

url = {http://eudml.org/doc/286807},

volume = {66},

year = {2016},

}

TY - JOUR

AU - Bart, Harm

AU - Ehrhardt, Torsten

AU - Silbermann, Bernd

TI - Rank decomposition in zero pattern matrix algebras

JO - Czechoslovak Mathematical Journal

PY - 2016

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 66

IS - 3

SP - 987

EP - 1005

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/286807

ER -

## References

top- Bart, H., Ehrhardt, T., Silbermann, B., Echelon type canonical forms in upper triangular matrix algebras, (to appear) in Oper. Theory, Adv. Appl.
- Bart, H., Ehrhardt, T., Silbermann, B., Sums of idempotents and logarithmic residues in zero pattern matrix algebras, Linear Algebra Appl. 498 (2016), 262-316. (2016) Zbl1334.15039MR3478563
- Bart, H., Ehrhardt, T., Silbermann, B., Sums of idempotents and logarithmic residues in matrix algebras, Operator Theory and Analysis. The M. A. Kaashoek Anniversary Volume (Bart, H. et al. eds.), Oper. Theory, Adv. Appl. 122 (2001), 139-168. (2001) Zbl1047.46037MR1846056
- Bart, H., Ehrhardt, T., Silbermann, B., 10.1007/BF01206410, Integral Equations Oper. Theory 19 (1994), 135-152. (1994) Zbl0811.46043MR1274555DOI10.1007/BF01206410
- Bart, H., Wagelmans, A. P. M., An integer programming problem and rank decomposition of block upper triangular matrices, Linear Algebra Appl. 305 (2000), 107-129. (2000) Zbl0951.15013MR1733797
- Birkhoff, G., Lattice Theory, Colloquium Publications Vol. 25 American Mathematical Society, Providence (1967). (1967) Zbl0153.02501MR0227053
- Davis, R. L., 10.1016/S0021-9800(70)80064-3, J. Comb. Theory 9 (1970), 257-260. (1970) Zbl0211.06401MR0268208DOI10.1016/S0021-9800(70)80064-3
- Fang, S.-C., Puthenpura, S., Linear Optimization and Extensions: Theory and Algorithms, Prentice-Hall, Englewood Cliffs (1993). (1993)
- Harary, F., Graph Theory, Addison-Wesley Series in Mathematics, Reading, Mass. Addison-Wesley Publishing Company (1969). (1969) Zbl0196.27202MR0256911
- Laffey, T. J., A structure theorem for some matrix algebras, Linear Algebra Appl. 162-164 (1992), 205-215. (1992) Zbl0758.16010MR1148400
- Papadimitriou, C. H., Steiglitz, K., Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs (1982). (1982) Zbl0503.90060MR0663728
- Schrijver, A., Theory of Linear and Integer Programming, Wiley-Interscience Series in Discrete Mathematics John Wiley & Sons, Chichester (1986). (1986) Zbl0665.90063MR0874114
- Szpilrajn, E., 10.4064/fm-16-1-386-389, Fundamenta Mathematicae 16 (1930), 386-389 French available at http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm16125.pdf. (1930) DOI10.4064/fm-16-1-386-389

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.