薛定宇教授大講堂（捲Ⅲ）：MATLAB線性代數運算（第2版）

目　

錄

CONTENTSCONTENTS

第 1章線性代數簡介 ······································· 1

1.1矩陣與線性方程組···································· 1

1.1.1表格的矩陣表示································ 1

1.1.2線性方程組的建立與求解 ························ 4

1.2線性代數發展簡介···································· 8

1.2.1線性代數數學理論 ······························ 8

1.2.2數值線性代數·································· 10本章習題 ················································ 12第 2章矩陣的表示與基本運算 ································ 14

2.1一般矩陣的輸入方法 ·································· 15

2.1.1矩陣的一般形式································ 15

2.1.2實矩陣的輸入·································· 15

2.1.3復矩陣的輸入·································· 16

2.1.4矩陣對稱性測試································ 16

2.2特殊矩陣的生成方法 ·································· 17

2.2.1零矩陣、幺矩陣及單位陣 ························· 17

2.2.2 Hankel矩陣 ··································· 18

2.2.3對角元素矩陣·································· 19

2.2.4 Hilbert矩陣及 Hilbert逆矩陣······················ 21

2.2.5相伴矩陣 ····································· 22

2.2.6 Wilkinson矩陣 ································· 23

2.2.7 Vandermonde矩陣 ······························ 23

2.2.8一些常用的測試矩陣 ···························· 24

2.3偽隨機數矩陣的生成 ·································· 25

2.3.1均勻分佈偽隨機數 ······························ 26

2.3.2隨機整數矩陣·································· 27

2.3.3測試矩陣生成·································· 28

? vi ?薛定宇教授大講堂（捲 III）：MATLAB線性代數運算（第 2版）

2.3.4正態分佈偽隨機數 ······························ 29

2.3.5常用分佈的偽隨機數 ···························· 31

2.4符號型矩陣的輸入方法 ································ 32

2.4.1特殊符號矩陣的輸入方法 ························ 32

2.4.2任意常數矩陣的輸入 ···························· 33

2.4.3任意矩陣函數的輸入 ···························· 34

2.5稀疏矩陣的輸入 ····································· 36

2.5.1一般稀疏矩陣的輸入與轉換······················· 36

2.5.2特殊稀疏矩陣的輸入 ···························· 39本章習題 ················································ 41

第 3章矩陣的基本運算······································ 46

3.1矩陣的轉置與旋轉···································· 47

3.1.1矩陣轉置與 Hermite轉置 ························· 47

3.1.2矩陣翻轉 ····································· 48

3.1.3矩陣的旋轉 ··································· 49

3.2矩陣的代數運算 ····································· 50

3.2.1矩陣的算術運算································ 50

3.2.2矩陣的乘方與開方 ······························ 53

3.2.3矩陣的點運算·································· 55

3.2.4 MATLAB的運算符 ····························· 57

3.2.5矩陣的 Kronecker乘積與 Kronecker和 ··············· 57

3.2.6復矩陣的處理·································· 60

3.2.7矩陣的絕對值與符號提取 ························ 61

3.3矩陣元素的非線性運算 ································ 61

3.3.1數據的取整與有理化運算 ························ 62

3.3.2超越函數計算命令 ······························ 63

3.3.3向量的排序、最大值與最小值······················ 66

3.3.4數據的均值、方差與標準差 ······················· 67

3.4矩陣函數的微積分運算 ································ 68

3.4.1矩陣函數的導數································ 68

3.4.2矩陣函數的積分································ 69

3.4.3向量函數的 Jacobi矩陣 ·························· 70

3.4.4 Hesse矩陣 ···································· 71本章習題 ················································ 72

目　錄 ? vii ?

第 4章矩陣基本分析 ······································· 75

4.1矩陣的行列式 ······································· 76

4.1.1行列式的定義與性質 ···························· 76

4.1.2矩陣行列式的計算方法 ·························· 77

4.1.3行列式計算問題的 MATLAB求解 ·················· 81

4.1.4任意階特殊矩陣的行列式計算····················· 84

4.1.5線性方程組的 Cramer法則 ······················· 85

4.1.6正矩陣與完全正矩陣 ···························· 87

4.2矩陣的簡單分析 ····································· 88

4.2.1矩陣的跡 ····································· 88

4.2.2線性無關與矩陣的秩 ···························· 89

4.2.3矩陣的範數 ··································· 91

4.2.4向量空間 ····································· 94

4.3逆矩陣與廣義逆矩陣 ·································· 96

4.3.1矩陣的逆矩陣·································· 96

4.3.2 MATLAB提供的矩陣求逆函數 ···················· 98

4.3.3簡化的行階梯型矩陣 ···························· 100

4.3.4逆矩陣的導函數································ 102

4.3.5矩陣的廣義逆矩陣 ······························ 104

4.4特徵多項式與特徵值 ·································· 107

4.4.1矩陣的特徵多項式 ······························ 107

4.4.2多項式方程的求根 ······························ 109

4.4.3一般矩陣的特徵值與特徵向量····················· 111

4.4.4矩陣的廣義特徵向量問題 ························ 115

4.4.5 Gershgorin圓盤與對角占優矩陣 ··················· 116

4.5矩陣多項式 ········································· 119

4.5.1矩陣多項式的求解 ······························ 119

4.5.2矩陣的最小多項式 ······························ 120

4.5.3符號多項式與數值多項式的轉換 ··················· 121本章習題 ················································ 123

第 5章矩陣的基本變換與分解 ································ 128

5.1相似變換與正交矩陣 ·································· 129

5.1.1相似變換 ····································· 129

5.1.2正交矩陣與正交基 ······························ 130

5.1.3 Schmidt正交化方法····························· 131

? viii ?薛定宇教授大講堂（捲 III）：MATLAB線性代數運算（第 2版）

5.2初等行變換 ········································· 132

5.2.1三種初等行變換方法 ···························· 132

5.2.2用初等行變換的方法求逆矩陣····················· 135

5.2.3主元素方法求逆矩陣 ···························· 136

5.3矩陣的三角分解 ····································· 137

5.3.1線性方程組的 Gauss消去法 ······················· 137

5.3.2一般矩陣的三角分解方法與實現 ··················· 138

5.3.3 MATLAB三角分解函數 ························· 139

5.4矩陣的 Cholesky分解·································· 141

5.4.1對稱矩陣的 Cholesky分解 ························ 141

5.4.2對稱矩陣的二次型表示 ·························· 142

5.4.3正定矩陣與正規矩陣 ···························· 143

5.4.4非正定矩陣的 Cholesky分解 ······················ 145

5.5相伴變換與 Jordan變換································ 145

5.5.1一般矩陣變換成相伴矩陣 ························ 146

5.5.2矩陣的對角化·································· 146

5.5.3矩陣的 Jordan變換 ····························· 147

5.5.4復特徵值矩陣的實 Jordan分解 ···················· 149

5.5.5正定矩陣的同時對角化 ·························· 151

5.6奇異值分解 ········································· 151

5.6.1奇異值與條件數································ 152

5.6.2長方形矩陣的奇異值分解 ························ 154

5.6.3基於奇異值分解的同時對角化····················· 155

5.7 Givens變換與 Householder變換 ························· 155

5.7.1二維坐標的旋轉變換 ···························· 156

5.7.2一般矩陣的 Givens變換 ·························· 158

5.7.3 Householder變換 ······························· 160本章習題 ················································ 161

第 6章矩陣方程求解 ······································· 164

6.1線性方程組 ········································· 165

6.1.1唯一解的求解·································· 166

6.1.2方程無窮解的求解與構造 ························ 170

6.1.3矛盾方程的求解································ 173

6.1.4線性方程解的幾何解釋 ·························· 174

目　錄 ? ix ?

6.2其他形式的簡單線性方程組 ···························· 175

6.2.1方程 XA = B的求解 ··························· 175

6.2.2方程 AXB = C的求解·························· 177

6.2.3基於 Kronecker乘積的方程解法···················· 179

6.2.4多項方程 AXB = C的求解 ······················ 179

6.3 Lyapunov方程······································· 180

6.3.1連續 Lyapunov方程 ····························· 180

6.3.2二階 Lyapunov方程的 Kronecker乘積表示 ··········· 182

6.3.3一般 Lyapunov方程的解析解······················ 182

6.3.4 Stein方程的求解 ······························· 183

6.3.5離散 Lyapunov方程 ····························· 184

6.4 Sylvester方程 ······································· 185

6.4.1 Sylvester方程的數學形式與數值解 ················· 185

6.4.2 Sylvester方程的解析求解 ························ 186

6.4.3含參數 Sylvester方程的解析解 ···················· 188

6.4.4多項 Sylvester方程的求解 ························ 189

6.4.5廣義 Sylvester方程······························ 190

6.5非線性矩陣方程 ····································· 191

6.5.1 Riccati代數方程 ······························· 191

6.5.2一般多解非線性矩陣方程的數值求解 ··············· 193

6.5.3變形 Riccati方程的求解·························· 197

6.5.4一般非線性矩陣方程的數值求解 ··················· 198

6.6多項式方程的求解···································· 199

6.6.1多項式互質 ··································· 199

6.6.2 Diophantine多項式方程·························· 200

6.6.3偽多項式方程求根 ······························ 202本章習題 ················································ 204

第 7章矩陣函數 ··········································· 208

7.1矩陣指數函數計算···································· 209

7.1.1矩陣函數的定義與性質 ·························· 209

7.1.2矩陣指數函數的運算 ···························· 210

7.1.3基於 Taylor級數的截斷算法 ······················ 211

7.1.4基於 Cayley–Hamilton定理的算法·················· 212

7.1.5 MATLAB的直接計算函數························ 213

7.1.6基於 Jordan變換的求解方法 ······················ 214

? x ?薛定宇教授大講堂（捲 III）：MATLAB線性代數運算（第 2版）

7.2矩陣的對數與平方根函數計算 ·························· 215

7.2.1矩陣的對數運算································ 215

7.2.2矩陣的平方根運算 ······························ 217

7.3矩陣的三角函數運算 ·································· 218

7.3.1矩陣的三角函數運算 ···························· 218

7.3.2基於 Taylor級數展開的矩陣三角函數計算 ··········· 218

7.3.3矩陣三角函數的解析求解 ························ 220

7.4一般矩陣函數的運算 ·································· 222

7.4.1冪零矩陣 ····································· 222

7.4.2基於 Jordan變換的矩陣函數運算 ·················· 223

7.4.3矩陣自定義函數的運算 ·························· 226

7.5矩陣的乘方運算 ····································· 227

7.5.1基於 Jordan變換的矩陣乘方運算 ·················· 227

7.5.2通用乘方函數的編寫 ···························· 228

7.5.3基於 z變換的矩陣乘方計算 ······················· 229

7.5.4計算矩陣乘方 kA ······························· 231本章習題 ················································ 231第 8章線性代數的應用······································ 233

8.1向量空間的幾何應用 ·································· 234

8.1.1向量及運算 ··································· 234

8.1.2直線方程 ····································· 237

8.1.3平面方程 ····································· 239

8.1.4最短距離的計算································ 242

8.1.5二次曲面方程·································· 244

8.2線性方程組的應用···································· 246

8.2.1電路網絡分析·································· 247

8.2.2結構平衡的分析方法 ···························· 252

8.2.3化學反應方程式配平 ···························· 252

8.3線性控制系統中的應用 ································ 255

8.3.1控制系統的模型轉換 ···························· 255

8.3.2線性系統的定性分析 ···························· 256

8.3.3多變量系統的傳輸零點 ·························· 258

8.3.4線性微分方程的直接求解 ························ 259

目　錄 ? xi ?

8.4數字圖像處理應用簡介 ································ 263

8.4.1圖像的讀入與顯示 ······························ 263

8.4.2矩陣的奇異值分解 ······························ 264

8.4.3圖像幾何尺寸變換與旋轉 ························ 266

8.4.4圖像增強 ····································· 267

8.5圖論與應用 ········································· 269

8.5.1有向圖的描述·································· 270

8.5.2 Dijkstra最短路徑算法及實現 ····················· 272

8.5.3控制系統方框圖化簡 ···························· 275

8.6差分方程求解 ······································· 278

8.6.1一般差分方程的解析解方法······················· 279

8.6.2線性時變差分方程的數值解方法 ··················· 280

8.6.3線性時不變差分方程的解法······················· 282

8.6.4一般非線性差分方程的數值解方法 ················· 283

8.6.5 Markov鏈的模擬 ······························· 284

8.7數據擬合與分析 ····································· 286

8.7.1線性回歸 ····································· 286

8.7.2多項式擬合 ··································· 288

8.7.3 Chebyshev多項式 ······························ 290

8.7.4 Bézier曲線 ···································· 292

8.7.5主成分分析 ··································· 294本章習題 ················································ 297參考文獻··················································· 303 MATLAB函數名索引 ········································· 305

術語索引··················································· 310