|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 , , Sylvester , Frobenius  in the nineteenth century, followed by the work Kreis  and Cecioni  in the early twentieth century. For more recent work on the pth roots or square roots of matrices, see the reference list of . Motivated by the use of stochastic matrices in the theory of Markov chain models, in  by exploiting the theory of functions, Highman and Lin have considered the question of under what conditions a given stochastic matrix has stochastic pth roots. Except for this and some other indirect results (see, for instance, ), 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 matrix to have a 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. 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 (V1; V2;E) such that
V1 consists of two vertices and G does not contain every possible arcs from V1 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.