摘要: | By a square root of a (square) matrix A we mean a matrix B that satisfies B2 = A . The study of square roots or pth roots of a general (real or complex) matrix can be traced back to the early work of Cayley [1] , [2], Sylvester [8], Frobenius [4] in the nineteenth century, followed by the work Kreis [7] and Cecioni [3] in the early twentieth century. For more recent work on the pth roots or square roots of matrices, see the reference list of [5]. Motivated by the use of stochastic matrices in the theory of Markov chain models, in [5] by exploiting the theory of functions, Highman and Lin have considered the quest ion of under what conditions a given stochastic matrix
has stochastic pth roots. Except for this and some other indirect results (see, for instance, [6]), there is no much literature addressed to (entrywise) nonnegative square roots or pth roots of a nonnegative matrix. Here we offer a study of the nonnegative square root problem, adopting mainly a graph-theeoretic approach. To start with, we settle completely the question of existence and uniqueness of nonnegative square roots for 2-by-2 nonnegative matrices. Then we connect the nonnegative square root problem with the square root problem for a digraph. 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 of 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 mat rix to have a square root is that its d igraph 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. For a digraph that is a path, a circuit or a disjoint union of paths and/or circuits, we characterize when the digraph has a square root. We also identify certain kinds of digraphs G, which necessarily have square roots , with the property that every nonnegative matrix with G as its digraph has a nonnegative square root; they include a circuit of odd length, the disjoint union of two circuits with the same odd length, a bigraph (VI, V2 , E) such that VI consists of two vertices and G does not contain every possible arcs from VI to V2 , etc. We characterize when a nonnegative generalized permutation matrix and a rank-one nonnegative matrix has a nonnegative square root. It is also found that a necessary condition for a symmetric nonnegative matrix A to have a symmetric nonnegative square root is that the eigenvalues of A are all nonnegative and the digraph of A has a symmetric square root. Some open questions will be mentioned.
This talk is based on a joint work with Peng-Rui Huang.
REFERENCES
[1] A. Cayley, A memoir on the theory of matr ices, P hil. Trans. Roy. Soc. London, 148 (1858), 17-37.
[2] A. Cayley, On the extraction of the square roots of a matrix of the third order, Proc. Roy. Soc. Edinburgh, 7 (1872), 675-682.
[3] F. Cecioni, Sopra alcune operazioni algebriche sulle matrici, Ann. Scnola Norm. Sup. Pisa Cl. Sci., 11 (1910), 1-141.
[4] G. Frobenius, Uber die cogredienten transformation der bil inearen fonnen, Sitzllngsber. Konigl. P ress. Akad. Wiss., (1!J86), 7-16.
[5] N.J. Higham and L.J. Lin, On pth roots of stochastic matrices, Linear Algebra App!., 435 (2011), 448-463.
[6] M. Marcus and H. Minc, Some results on doubly stochastic matrices, Proc. Amer. Math. Soc., 69 (1962), 571-57!J.
[7] H. Kreis, Auflosung der gleichung xm = A, Vuschr . Natnrforsch . Ges. Ziirich, 53 (1908), 366-376.
[8] J.J. Sylvester, Sur les puissances et les racines de substitutions lineaires, Comptes Rendus, 94 (1882), 55-59. |