Distance matrices perturbed by Laplacians

Ramamurthy, Balaji, Bapat, Ravindra Bhalchandra, and Goel, Shivani. "Distance matrices perturbed by Laplacians." Applications of Mathematics 65.5 (2020): 599-607. <http://eudml.org/doc/296973>.

@article{Ramamurthy2020,
abstract = {Let $T$ be a tree with $n$ vertices. To each edge of $T$ we assign a weight which is a positive definite matrix of some fixed order, say, $s$. Let $D_\{ij\}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$. We now say that $D_\{ij\}$ is the distance between $i$ and $j$. Define $D:=[D_\{ij\}]$, where $D_\{ii\}$ is the $s \times s$ null matrix and for $i \ne j$, $D_\{ij\}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order $s$. If $i$ and $j$ are adjacent, then define $L_\{ij\}:=-W_\{ij\}^\{-1\}$, where $W_\{ij\}$ is the weight of the edge $(i,j)$. Define $L_\{ii\}:=\sum _\{i \ne j,j=1\}^\{n\}W_\{ij\}^\{-1\}$. The Laplacian of $G$ is now the $ns \times ns$ block matrix $L:=[L_\{ij\}]$. In this paper, we first note that $D^\{-1\}-L$ is always nonsingular and then we prove that $D$ and its perturbation $(D^\{-1\}-L)^\{-1\}$ have many interesting properties in common.},
author = {Ramamurthy, Balaji, Bapat, Ravindra Bhalchandra, Goel, Shivani},
journal = {Applications of Mathematics},
keywords = {tree; Laplacian matrix; inertia; Haynsworth formula},
language = {eng},
number = {5},
pages = {599-607},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Distance matrices perturbed by Laplacians},
url = {http://eudml.org/doc/296973},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Ramamurthy, Balaji
AU - Bapat, Ravindra Bhalchandra
AU - Goel, Shivani
TI - Distance matrices perturbed by Laplacians
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 5
SP - 599
EP - 607
AB - Let $T$ be a tree with $n$ vertices. To each edge of $T$ we assign a weight which is a positive definite matrix of some fixed order, say, $s$. Let $D_{ij}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$. We now say that $D_{ij}$ is the distance between $i$ and $j$. Define $D:=[D_{ij}]$, where $D_{ii}$ is the $s \times s$ null matrix and for $i \ne j$, $D_{ij}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order $s$. If $i$ and $j$ are adjacent, then define $L_{ij}:=-W_{ij}^{-1}$, where $W_{ij}$ is the weight of the edge $(i,j)$. Define $L_{ii}:=\sum _{i \ne j,j=1}^{n}W_{ij}^{-1}$. The Laplacian of $G$ is now the $ns \times ns$ block matrix $L:=[L_{ij}]$. In this paper, we first note that $D^{-1}-L$ is always nonsingular and then we prove that $D$ and its perturbation $(D^{-1}-L)^{-1}$ have many interesting properties in common.
LA - eng
KW - tree; Laplacian matrix; inertia; Haynsworth formula
UR - http://eudml.org/doc/296973
ER -