On the core-nilpotent decomposition of trees

JAUME, DANIEL A.; SOTA, RODRIGO

LINEAR ALGEBRA AND ITS APPLICATIONS

ELSEVIER SCIENCE INC

Año: 2019 vol. 563 p. 207 - 214

0024-3795

In this work we show, through the null decomposition of trees given by Jaume and Molina (2018) [3], that the core-nilpotent decomposition of A(T), the adjacency matrix of a tree T, can be obtained directly from the tree itself. In other words, we give two invertible matrices Q and C, expressed in terms of the adjacency relations of T, such that Q−1A(T)Q is a 2×2 blocks diagonal matrix, whose first block is C (a r×r matrix such that rk(C)=rk(A(T))=r) and whose second block is a zero matrix. Using the results given by Jaume et al. (2018) [4], the core-nilpotent decomposition of trees can be obtained in linear time.