### Abstract

Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi's conditions are sharp. Improving Varga's results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular.

Original language | English (US) |
---|---|

Pages (from-to) | 14-19 |

Number of pages | 6 |

Journal | Linear Algebra and Its Applications |

Volume | 428 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2008 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Keywords

- Assignment problem
- Digraph
- Duality
- Matrix singularity
- Scwaltcy
- Transversal

### Cite this

*Linear Algebra and Its Applications*,

*428*(1), 14-19. https://doi.org/10.1016/j.laa.2007.10.003

}

*Linear Algebra and Its Applications*, vol. 428, no. 1, pp. 14-19. https://doi.org/10.1016/j.laa.2007.10.003

**Geršgorin variations III : On a theme of Brualdi and Varga.** / Boros, Endre; Brualdi, Richard A.; Crama, Yves; Hoffman, A. J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Geršgorin variations III

T2 - On a theme of Brualdi and Varga

AU - Boros, Endre

AU - Brualdi, Richard A.

AU - Crama, Yves

AU - Hoffman, A. J.

PY - 2008/1/1

Y1 - 2008/1/1

N2 - Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi's conditions are sharp. Improving Varga's results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular.

AB - Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi's conditions are sharp. Improving Varga's results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular.

KW - Assignment problem

KW - Digraph

KW - Duality

KW - Matrix singularity

KW - Scwaltcy

KW - Transversal

UR - http://www.scopus.com/inward/record.url?scp=36048986362&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36048986362&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2007.10.003

DO - 10.1016/j.laa.2007.10.003

M3 - Article

AN - SCOPUS:36048986362

VL - 428

SP - 14

EP - 19

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1

ER -