Skip to main content
SpringerLink
Log in

When does the inverse have the same sign pattern as the transpose?

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

By a sign pattern (matrix) we mean an array whose entries are from the set {+, −, 0}. The sign patterns A for which every real matrix with sign pattern A has the property that its inverse has sign pattern A T are characterized. Sign patterns A for which some real matrix with sign pattern A has that property are investigated. Some fundamental results as well as constructions concerning such sign pattern matrices are provided. The relation between these sign patterns and the sign patterns of orthogonal matrices is examined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beasley, L. B., R. A. Brualdi, and B. L. Shader: Combinatorial orthogonality. Combinatorial and Graph-Theoretic Problems in Linear Algebra, IMA Vol. Math. Appl. 50 (R. Brualdi, S. Friedland, and V. Klee, eds.). Springer-Verlag, 1993, pp. 207–218.

  2. Brualdi, R. A., K. L. Chavey, and B. L. Shader: Bipartite graphs and inverse sign patterns of strong sign-nonsingular matrices. Journal of Combinatorial Theory (B) 62 (1994), 133–152.

    Google Scholar 

  3. Beasley, L. B. and D. J. Scully: Linear operators which preserve combinatorial orthogonality. Linear Algebra and its Applications 201 (1994), 171–180.

    Google Scholar 

  4. Eschenbach, C. A., F. J. Hall, and Z. Li: Some sign patterns that allow a real Inverse-Pair B and B −1. Linear Algebra and its Applications 252 (1997), 299–321.

    Google Scholar 

  5. Fiedler, M., Editor: Proceedings: Theory of Graphs and Its Applications. Publishing House of Czechoslovakia Academy of Sciences, Prague, 1964.

    Google Scholar 

  6. Horn, R. A. and C. R. Johnson: Matrix Analysis. Cambridge, 1985.

  7. Horn, R. A. and C. R. Johnson: Topics in Matrix Analysis. Cambridge, 1991.

  8. Jeng, J-H.: On sign patterns of orthogonal matrices. Master's Thesis, Tam Chiang Univ., 1984.

  9. Johnson, C. R., F. T. Leighton, and H. A. Robinson: Sign patterns of Inverse-Positive matrices. Linear Algebra and its Applications 24 (1979), 75–83.

    Google Scholar 

  10. Thomassen, C.: Sign-nonsingular matrices and even cycles in directed graphs. Linear Algebra and its Applications 75 (1986), 27–41.

    Google Scholar 

  11. Waters, C.: Sign patterns that allow orthogonality. Linear Algebra and its Applications 235 (1996), 1–13.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Georgia State University, Atlanta, GA, 30303, USA

    Carolyn A. Eschenbach, Frank J. Hall, Deborah L. Harrell & Zhongshan Li

Authors
  1. Carolyn A. Eschenbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank J. Hall
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Deborah L. Harrell
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhongshan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eschenbach, C.A., Hall, F.J., Harrell, D.L. et al. When does the inverse have the same sign pattern as the transpose?. Czechoslovak Mathematical Journal 49, 255–275 (1999). https://doi.org/10.1023/A:1022496101277

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022496101277

Keywords

Search

Navigation