| Title:
|
Distance matrices perturbed by Laplacians (English) |
| Author:
|
Ramamurthy, Balaji |
| Author:
|
Bapat, Ravindra Bhalchandra |
| Author:
|
Goel, Shivani |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
65 |
| Issue:
|
5 |
| Year:
|
2020 |
| Pages:
|
599-607 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $i \neq j$, $D_{ij}$ 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 $L_{ii}:=\sum _{i \neq j,j=1}^{n}W_{ij}^{-1}$. 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. (English) |
| Keyword:
|
tree |
| Keyword:
|
Laplacian matrix |
| Keyword:
|
inertia |
| Keyword:
|
Haynsworth formula |
| MSC:
|
05C50 |
| MSC:
|
15B48 |
| idZBL:
|
07285947 |
| idMR:
|
MR4160783 |
| DOI:
|
10.21136/AM.2020.0362-19 |
| . |
| Date available:
|
2020-09-23T13:48:25Z |
| Last updated:
|
2022-11-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148367 |
| . |
| Reference:
|
[1] Balaji, R., Bapat, R. B.: Block distance matrices.Electron. J. Linear Algebra 16 (2007), 435-443. Zbl 1148.15016, MR 2365897, 10.13001/1081-3810.1213 |
| Reference:
|
[2] Bapat, R. B.: Determinant of the distance matrix of a tree with matrix weights.Linear Algebra Appl. 416 (2006), 2-7. Zbl 1108.15006, MR 2232916, 10.1016/j.laa.2005.02.022 |
| Reference:
|
[3] Bapat, R., Kirkland, S. J., Neumann, M.: On distance matrices and Laplacians.Linear Algebra Appl. 401 (2005), 193-209. Zbl 1064.05097, MR 2133282, 10.1016/j.laa.2004.05.011 |
| Reference:
|
[4] Fiedler, M.: Matrices and Graphs in Geometry.Encyclopedia of Mathematics and Its Applications 139. Cambridge University Press, Cambridge (2011). Zbl 1225.51017, MR 2761077, 10.1017/CBO9780511973611 |
| Reference:
|
[5] Fiedler, M., Markham, T. L.: Completing a matrix when certain entries of its inverse are specified.Linear Algebra Appl. 74 (1986), 225-237. Zbl 0592.15002, MR 0822149, 10.1016/0024-3795(86)90125-4 |
| . |
