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

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...

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Bibliographic Details
Published in:Linear algebra and its applications 2023-09, Vol.673, p.200-219
Main Authors: Kager, Wouter, Storm, Pieter Jacob
Format: Article
Language:English
Subjects:
Determinant sign
Directed graph
Matrix class
P-matrix
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2023.05.013