Sufficient conditions to be exceptional

Charles R. Johnson; Robert B. Reams

Special Matrices (2016)

  • Volume: 4, Issue: 1, page 67-72
  • ISSN: 2300-7451

Abstract

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A copositive matrix A is said to be exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. We show that with certain assumptions on A−1, especially on the diagonal entries, we can guarantee that a copositive matrix A is exceptional. We also show that the only 5-by-5 exceptional matrix with a hollow nonnegative inverse is the Horn matrix (up to positive diagonal congruence and permutation similarity).

How to cite

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Charles R. Johnson, and Robert B. Reams. "Sufficient conditions to be exceptional." Special Matrices 4.1 (2016): 67-72. <http://eudml.org/doc/276827>.

@article{CharlesR2016,
abstract = {A copositive matrix A is said to be exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. We show that with certain assumptions on A−1, especially on the diagonal entries, we can guarantee that a copositive matrix A is exceptional. We also show that the only 5-by-5 exceptional matrix with a hollow nonnegative inverse is the Horn matrix (up to positive diagonal congruence and permutation similarity).},
author = {Charles R. Johnson, Robert B. Reams},
journal = {Special Matrices},
keywords = {copositive matrix; positive semidefinite; nonnegative matrix; exceptional copositive matrix; irreducible matrix},
language = {eng},
number = {1},
pages = {67-72},
title = {Sufficient conditions to be exceptional},
url = {http://eudml.org/doc/276827},
volume = {4},
year = {2016},
}

TY - JOUR
AU - Charles R. Johnson
AU - Robert B. Reams
TI - Sufficient conditions to be exceptional
JO - Special Matrices
PY - 2016
VL - 4
IS - 1
SP - 67
EP - 72
AB - A copositive matrix A is said to be exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. We show that with certain assumptions on A−1, especially on the diagonal entries, we can guarantee that a copositive matrix A is exceptional. We also show that the only 5-by-5 exceptional matrix with a hollow nonnegative inverse is the Horn matrix (up to positive diagonal congruence and permutation similarity).
LA - eng
KW - copositive matrix; positive semidefinite; nonnegative matrix; exceptional copositive matrix; irreducible matrix
UR - http://eudml.org/doc/276827
ER -

References

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  8. [8] M. Hall, Combinatorial theory, Blaisdell/Ginn, 1967. 
  9. [9] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, 1985. Zbl0576.15001
  10. [10] M. Hall and M. Newman, Copositive and completely positive quadratic forms, Proc. Camb. Phil. Soc.59 (1963) 329–339.[Crossref] Zbl0124.25302
  11. [11] A. J. Hoffman and F. Pereira, On copositive matrices with −1, 0, 1 entries, Journal of Combinatorial Theory (A)14 (1973) 302–309. Zbl0273.15019
  12. [12] C. R. Johnson and R. Reams, Constructing copositive matrices from interior matrices, Electronic Journal of Linear Algebra17 (2008) 9–20. Zbl1143.15023
  13. [13] C. R. Johnson and R. Reams, Spectral theory of copositive matrices, Linear Algebra and its Applications395 (2005) 275–281. Zbl1064.15007
  14. [14] H. Minc, Nonnegative Matrices, Wiley, New York, 1988. 
  15. [15] H. Väliaho, Criteria for copositive matrices, Linear Algebra and its Applications81 (1986) 19–34.[Crossref] 

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