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 [11], Frobenius [6] in the nineteenth century, followed by the work Kreis [10] 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 [7]. Motivated by the use of stochastic matrices in the theory of Markov chain models, in [7] 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, [8]), 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. |