Nonnegative square roots of matrices

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• 作者：Bit-Shun Tam ; ¹ ; ^{bsm01@mail.tku.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Peng-Ruei Huang ^{143422@mail.tku.edu.tw" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15B48 ; 05C20 ; 05C25 ; 05C76
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：498
• 期：Complete
• 页码：404-440
• 全文大小：621 K

文摘

By the square root of a (square) matrix A we mean a matrix B   that satisfies class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006746&_mathId=si1.gif&_user=111111111&_pii=S0024379515006746&_rdoc=1&_issn=00243795&md5=c188df540aff315d36dc7380cc70e91b" title="Click to view the MathML source">B²=Aclass="mathContainer hidden">class="mathCode">$B^2=A$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006746&_mathId=si2.gif&_user=111111111&_pii=S0024379515006746&_rdoc=1&_issn=00243795&md5=64650d9a65ad28926004e1e1ad9243e1" title="Click to view the MathML source">H²class="mathContainer hidden">class="mathCode">$H^2$, we mean the digraph with the same vertex set as H   such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006746&_mathId=si3.gif&_user=111111111&_pii=S0024379515006746&_rdoc=1&_issn=00243795&md5=b2ebfd1c1f7fb3323137be24e9ec060a" title="Click to view the MathML source">(i,j)class="mathContainer hidden">class="mathCode">$(i,j)$ is an arc if there is a vertex k   such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006746&_mathId=si4.gif&_user=111111111&_pii=S0024379515006746&_rdoc=1&_issn=00243795&md5=99aa3f6930c942304904f3dfb5bf051d" title="Click to view the MathML source">(i,k)class="mathContainer hidden">class="mathCode">$(i,k)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006746&_mathId=si5.gif&_user=111111111&_pii=S0024379515006746&_rdoc=1&_issn=00243795&md5=b343b72e8601924b17be6c0aa282f235" title="Click to view the MathML source">(k,j)class="mathContainer hidden">class="mathCode">$(k,j)$ are both arcs in H. We call a digraph H a square root of a digraph G   if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379515006746&_mathId=si6.gif&_user=111111111&_pii=S0024379515006746&_rdoc=1&_issn=00243795&md5=ee1931b360a830034793516cbdd8c238" title="Click to view the MathML source">H²=Gclass="mathContainer hidden">class="mathCode">$H^2=G$. 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.