一般方陣的平方根問題，在許多現有的文獻中已經可以找到答案。
但並無太多文獻上的結果提到非負矩陣之非負平方根的存在性問題。本文主要探討，非負矩陣在甚麼條件下擁有非負平方根。首先討論並且完整刻劃2階非負矩陣之存在性與唯一性問題。我們也得出一些結果知道在甚麼情況下，一個有向圖會有平方根，從而可以藉由非負矩陣所伴隨的有向圖，判斷非負矩陣是否有非負平方根。本文主要探討的有向圖則是路徑、迴圈、置換有向圖以及偶圖。此外，我們也得出秩1非負矩陣擁有非負p次方根與單項非負矩陣之非負平方根存在性的充分且必要條件及考慮對稱非負方陣擁有對稱非負平方根的存在性問題。

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.

Preface                                                v
Acknowledgement                                      vii
Notation                                              ix
1 Introduction                                         1
1.1 Historical Background . . . . . . .. . . . . . . . 1
1.2 Preliminary Results . . . . .. . . . . . . . . . . 2
1.3 Square Roots for Complex and Real Matrices . . . . 4

2 The Two-by-Two Case                                  6
2.1 Main Theorem . . . . . . . . . . . . . . . . . . . 6
2.2 Remarks and Examples . . . .  . . . . . . . . . . 11

3 Square Roots of Digraphs                            15
3.1 De nitions and Notations . .  . . . . . . .  . .. 15
3.2 Directed Paths and Circuits . . . . . . . .  . .. 17
3.3 Bigraphs . . . . . . . . . . . . . . . . . . . .. 26

4 Special Classes of Nonnegative Matrices             34
4.1 Symmetric Matrices . . . . . . . . . .  . .  . .. 34
4.2 Rank-One Matrices . . . . . . . . . . . . . . . . 40

5 Final Remarks and Open Questions                    46
5.1 Complete Positivity . . . . . . . . . . . . . . . 46
5.2 Necessary Conditions Arising from the 
Nonnegative Inverse Eigenvalue Problem  .  . .. .   . 49
5.3 Max Algebra . . . . . . . . . . . . . . .  . .. . 51
5.4 Primes in the Semigroup of Nonnegative Matrices . 53
5.5 Soules Basis . . .. . . . . . . . . . . . . . . . 54
5.6 The Equations $Bj = Aj . . . AmA1 . . .  A_{j-1}, 
1 leq m leq  j$ . . . . . . . . . . . . . . . . . . . 56
5.7 Directions for Further Research . . . . . . . . . 57

References                                            59

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