Nonnegative square roots of matrices

doi:10.1016/j.laa.2015.11.011

機構典藏 > College of Sciences > Graduate Institute & Department of Mathematics > Journal Article > Item 987654321/106937

Please use this identifier to cite or link to this item: https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/106937

Title:	Nonnegative square roots of matrices
Authors:	Bit-ShunTam;Peng-RueiHuang
Keywords:	Nonnegative matrix;Nonnegative square root;Square root of digraph;Permutation digraph;Bigraph;Monomial matrices;Nilpotent matrix;Rank-one matrix
Date:	2016-06-01
Issue Date:	2016-08-15
Publisher:	Elsevier Inc.
Abstract:	By the square root of a (square) matrix A we mean a matrix B that satisfies B2=A. In this paper, we begin a study of the (entrywise) nonnegative square roots of nonnegative matrices, adopting mainly a graph-theoretic approach. To start with, we settle completely the question of existence and uniqueness of nonnegative square roots for 2-by-2 nonnegative matrices. By the square of a digraph H , denoted by H2, we mean the digraph with the same vertex set as H such that (i,j) is an arc if there is a vertex k such that (i,k) and (k,j) are both arcs in H. We call a digraph H a square root of a digraph G if H2=G. It is observed that a necessary condition for a nonnegative matrix to have a nonnegative square root is that its digraph has a square root, and also that a digraph G has a square root if and only if there exists a nonnegative matrix A with digraph G such that A has a nonnegative square root. We consider when or whether certain kinds of digraphs (including digraphs that are disjoint union of directed paths and circuits, permutation digraphs or a special kind of bigraphs) have square roots. We also consider when certain kinds of nonnegative matrices (including monomial matrices, rank-one matrices and nilpotent matrices with index two) have nonnegative square roots. A known characterization of loopless digraphs to have square roots, due to F. Escalantge, L. Montejano, and T. Rojano, is extended (and amended) to digraphs possibly with loops. Some natural open questions are also posed.
Relation:	Linear Algebra and its Applications 498, pp.404-440
DOI:	10.1016/j.laa.2015.11.011
Appears in Collections:	[Graduate Institute & Department of Mathematics] Journal Article

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