#### Title

#### Campus Units

Mathematics

#### Document Type

Article

#### Publication Date

4-2010

#### Journal or Book Title

Linear Algebra and its Applications

#### Volume

432

#### First Page

19961

#### Last Page

1974

#### DOI

10.1016/j.laa.2009.05.003

#### Abstract

A graph describes the zero–nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303–316, C.R. Johnson, A. Leal Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999) 139–144], trees allowing loops [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness. Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl. 418 (2006) 389–415], and directed trees allowing loops [F. Barioli, S. Fallat, D. Hershkowitz, H.T. Hall, L. Hogben, H. van der Holst, B. Shader, On the minimum rank of not necessarily symmetric matrices: a preliminary study, Electron. J. Linear Algebra 18 (2000) 126–145]. We survey these results from a unified perspective and solve the minimum rank problem for simple directed trees.

#### Rights

This manuscript version is made available under the CCBY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

#### Copyright Owner

Elsevier Inc.

#### Copyright Date

2010

#### Language

en

#### File Format

application/pdf

#### Recommended Citation

Hogben, Leslie, "Minimum Rank Problems" (2010). *Mathematics Publications*. 89.

https://lib.dr.iastate.edu/math_pubs/89

## Comments

This is a manuscript of an article from

Linear Algebra and its Applications432 (2010): 1961, doi:10.1016/j.laa.2009.05.003. Posted with permission.