並列摘要
For a complex square matrix, there are many references on the question of the existence of a square root. However, not much is known about the question of existence of entrywise nonnegative square roots for an entrywise square nonnegative matrix. The purpose of this thesis is to address the question of when a nonnegative matrix has a nonnegative square root. We settle the existence and uniqueness question for $2 times 2$ nonnegative matrices. We relate the nonnegative square root problem for nonnegative matrices to the square root problem for digraphs, and focus on nonnegative matrices whose digraphs are paths, circuits, permutation digraphs or bigraphs. Moreover, we characterize rank-one nonnegative matrices that have nonnegative $p$th root, and nonnegative square roots of nonnegative monomial matrices, and also treat the question of when a symmetric nonnegative matrix has a symmetric nonnegative square root.