On the class of matrices with rows that weakly decrease cyclicly from the diagonal

Wouter Kager; Pieter Jacob Storm

doi:10.1016/j.laa.2023.05.013

On the class of matrices with rows that weakly decrease cyclicly from the diagonal

Wouter Kager^*
, Pieter Jacob Storm

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

We consider n×n real-valued matrices A=(a_ij) satisfying a_ii≥a_i,i+1≥…≥a_in≥a_i1≥…≥a_i,i−1 for i=1,…,n. With such a matrix A we associate a directed graph G(A). We prove that the solutions to the system A^⊤x=λe, with λ∈R and e the vector of all ones, are linear combinations of ‘fundamental’ solutions to A^⊤x=e and vectors in ker⁡A^⊤, each of which is associated with a closed strongly connected component (SCC) of G(A). This allows us to characterize the sign of det⁡A in terms of the number of closed SCCs and the solutions to A^⊤x=e. In addition, we provide conditions for A to be a P-matrix.

Original language	English
Pages (from-to)	200-219
Number of pages	20
Journal	Linear Algebra and its Applications
Volume	673
Early online date	16 May 2023
DOIs	https://doi.org/10.1016/j.laa.2023.05.013
Publication status	Published - 15 Sept 2023

Bibliographical note

Funding Information:
The authors are grateful to Universitas Gadjah Mada for partially funding this research through the RTA program.

Publisher Copyright:
© 2023 The Author(s)

Funding

The authors are grateful to Universitas Gadjah Mada for partially funding this research through the RTA program.

Keywords

Determinant sign
Directed graph
Matrix class
P-matrix

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10.1016/j.laa.2023.05.013

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title = "On the class of matrices with rows that weakly decrease cyclicly from the diagonal",

abstract = "We consider n×n real-valued matrices A=(aij) satisfying aii≥ai,i+1≥…≥ain≥ai1≥…≥ai,i−1 for i=1,…,n. With such a matrix A we associate a directed graph G(A). We prove that the solutions to the system A⊤x=λe, with λ∈R and e the vector of all ones, are linear combinations of {\textquoteleft}fundamental{\textquoteright} solutions to A⊤x=e and vectors in ker⁡A⊤, each of which is associated with a closed strongly connected component (SCC) of G(A). This allows us to characterize the sign of det⁡A in terms of the number of closed SCCs and the solutions to A⊤x=e. In addition, we provide conditions for A to be a P-matrix.",

keywords = "Determinant sign, Directed graph, Matrix class, P-matrix",

author = "Wouter Kager and Storm, \{Pieter Jacob\}",

note = "Funding Information: The authors are grateful to Universitas Gadjah Mada for partially funding this research through the RTA program. Publisher Copyright: {\textcopyright} 2023 The Author(s)",

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doi = "10.1016/j.laa.2023.05.013",

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T1 - On the class of matrices with rows that weakly decrease cyclicly from the diagonal

AU - Kager, Wouter

AU - Storm, Pieter Jacob

PY - 2023/9/15

Y1 - 2023/9/15

N2 - We consider n×n real-valued matrices A=(aij) satisfying aii≥ai,i+1≥…≥ain≥ai1≥…≥ai,i−1 for i=1,…,n. With such a matrix A we associate a directed graph G(A). We prove that the solutions to the system A⊤x=λe, with λ∈R and e the vector of all ones, are linear combinations of ‘fundamental’ solutions to A⊤x=e and vectors in ker⁡A⊤, each of which is associated with a closed strongly connected component (SCC) of G(A). This allows us to characterize the sign of det⁡A in terms of the number of closed SCCs and the solutions to A⊤x=e. In addition, we provide conditions for A to be a P-matrix.

AB - We consider n×n real-valued matrices A=(aij) satisfying aii≥ai,i+1≥…≥ain≥ai1≥…≥ai,i−1 for i=1,…,n. With such a matrix A we associate a directed graph G(A). We prove that the solutions to the system A⊤x=λe, with λ∈R and e the vector of all ones, are linear combinations of ‘fundamental’ solutions to A⊤x=e and vectors in ker⁡A⊤, each of which is associated with a closed strongly connected component (SCC) of G(A). This allows us to characterize the sign of det⁡A in terms of the number of closed SCCs and the solutions to A⊤x=e. In addition, we provide conditions for A to be a P-matrix.

KW - Determinant sign

KW - Directed graph

KW - Matrix class

KW - P-matrix

UR - https://www.scopus.com/pages/publications/85162198241

UR - https://www.scopus.com/pages/publications/85162198241#tab=citedBy

U2 - 10.1016/j.laa.2023.05.013

DO - 10.1016/j.laa.2023.05.013

M3 - Article

AN - SCOPUS:85162198241

SN - 0024-3795

VL - 673

SP - 200

EP - 219

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

ER -

On the class of matrices with rows that weakly decrease cyclicly from the diagonal

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Keywords

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