# 矩陣

${\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}$

m-by-n matrix」的各地常用別名

「橫向的一條線（row）」的各地常用別名

「縱向的一條線（column）」的各地常用別名

Matrix mxn (m row vectors x n column vectors)
${\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}$

## 定義

${\displaystyle \mathbf {A} ={\begin{bmatrix}9&13&5\\1&11&7\\3&9&2\\6&0&7\end{bmatrix}}}$

### 標記

${\displaystyle \mathbf {B} ={\begin{bmatrix}3&5&7\\4&6&8\end{bmatrix}}}$

## 矩陣的基本運算

(A ± B)i,j ${\displaystyle =}$ Ai,j ± Bi,j

${\displaystyle {\begin{bmatrix}1&3&1\\1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\end{bmatrix}}={\begin{bmatrix}1+0&3+0&1+5\\1+7&0+5&0+0\end{bmatrix}}={\begin{bmatrix}1&3&6\\8&5&0\end{bmatrix}}}$

(cA)i,j = c · Ai,j.
${\displaystyle 2\cdot {\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot (-3)\\2\cdot 4&2\cdot (-2)&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}}$

(AT)i,j = Aj,i.
${\displaystyle {\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{T}={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}}$

(A + B)T ${\displaystyle =}$ AT + BT
c(A + B) ${\displaystyle =}$ cA + cB

c(AT) ${\displaystyle =}$ (cA)T.

## 矩陣乘法

${\displaystyle [\mathbf {AB} ]_{i,j}=A_{i,1}B_{1,j}+A_{i,2}B_{2,j}+\cdots +A_{i,n}B_{n,j}=\sum _{r=1}^{n}A_{i,r}B_{r,j}}$

${\displaystyle {\begin{bmatrix}1&0&2\\-1&3&1\\\end{bmatrix}}\times {\begin{bmatrix}3&1\\2&1\\1&0\end{bmatrix}}={\begin{bmatrix}(1\times 3+0\times 2+2\times 1)&(1\times 1+0\times 1+2\times 0)\\(-1\times 3+3\times 2+1\times 1)&(-1\times 1+3\times 1+1\times 0)\\\end{bmatrix}}={\begin{bmatrix}5&1\\4&2\\\end{bmatrix}}}$

• 結合律：(AB)C ${\displaystyle =}$ A(BC),
• 左分配律：(A + B)C ${\displaystyle =}$ AC + BC,
• 右分配律：C(A + B) ${\displaystyle =}$ CA + CB.

c(AB) ${\displaystyle =}$ (cA)B ${\displaystyle =}$ A(cB)
(AB)T ${\displaystyle =}$ BTAT

${\displaystyle {\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}{\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}={\begin{bmatrix}0&1\\0&3\\\end{bmatrix}},\qquad \quad {\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\begin{bmatrix}1&2\\3&4\\\end{bmatrix}}={\begin{bmatrix}3&4\\0&0\\\end{bmatrix}}}$

### 線性方程組

${\displaystyle {\begin{cases}a_{1,1}x_{1}+a_{1,2}x_{2}+\cdots +a_{1,n}x_{n}=b_{1}\\a_{2,1}x_{1}+a_{2,2}x_{2}+\cdots +a_{2,n}x_{n}=b_{2}\\\vdots \quad \quad \quad \vdots \\a_{m,1}x_{1}+a_{m,2}x_{2}+\cdots +a_{m,n}x_{n}=b_{m}\end{cases}}}$

${\displaystyle \mathbf {A} \mathbf {x} =\mathbf {b} }$

${\displaystyle \mathbf {A} ={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\cdots &a_{m,n}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}$

### 線性變換

 推移， 幅度m=1.25. 水平鏡射變換 「擠壓」變換， 壓縮程度r=3/2 伸縮，3/2倍 旋轉，左轉30° ${\displaystyle {\begin{bmatrix}1&1.25\\0&1\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}-1&0\\0&1\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {2}{3}}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {3}{2}}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}\cos({\frac {\pi }{6}})&-\sin({\frac {\pi }{6}})\\\sin({\frac {\pi }{6}})&\cos({\frac {\pi }{6}})\end{bmatrix}}}$

(gf)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x

## 方塊矩陣

AB ${\displaystyle =}$ In

${\displaystyle {\begin{bmatrix}d_{11}&0&0\\0&d_{22}&0\\0&0&d_{33}\\\end{bmatrix}}}$（對角矩陣），${\displaystyle {\begin{bmatrix}l_{11}&0&0\\l_{21}&l_{22}&0\\l_{31}&l_{32}&l_{33}\\\end{bmatrix}}}$（下三角矩陣）和${\displaystyle {\begin{bmatrix}u_{11}&u_{12}&u_{13}\\0&u_{22}&u_{23}\\0&0&u_{33}\\\end{bmatrix}}}$（上三角矩陣）。

### 行列式

R2裏的一個線性變換f將藍色圖形變成綠色圖形，面積不變，而順時針排布的向量x1和x2的變成了逆時針排布。對應的矩陣行列式是-1.

2×2矩陣的行列式是

${\displaystyle \det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}=ad-bc}$

3×3矩陣的行列式由6項組成。更高維矩陣的行列式則可以使用萊布尼茲公式寫出[27]，或使用拉普拉斯展開由低一維的矩陣行列式遞推得出[28]

### 特徵值與特徵向量

n×n的方塊矩陣A的一個特徵值和對應特徵向量是滿足

${\displaystyle \mathbf {Av} =\lambda \mathbf {v} }$[32]的純量${\displaystyle \lambda }$以及非零向量${\displaystyle \mathbf {v} }$。特徵值和特徵向量的概念對研究線性變換很有幫助。一個線性變換可以通過它對應的矩陣在向量上的作用來可視化。一般來說，一個向量在經過映射之後可以變為任何可能的向量，而特徵向量具有更好的性質[33]。假設在給定的基底下，一個線性變換對應著某個矩陣A，如果一個向量x可以寫成矩陣的幾個特徵向量的線性組合：
${\displaystyle \mathbf {x} =c_{1}\mathbf {x} _{\lambda _{1}}+c_{2}\mathbf {x} _{\lambda _{2}}+\cdots +c_{k}\mathbf {x} _{\lambda _{k}}}$

${\displaystyle \mathbf {Ax} =c_{1}\lambda _{1}\mathbf {x} _{\lambda _{1}}+c_{2}\lambda _{2}\mathbf {x} _{\lambda _{2}}+\cdots +c_{k}\lambda _{k}\mathbf {x} _{\lambda _{k}}}$

${\displaystyle \det(\lambda {\mathsf {I}}_{n}-\mathbf {A} )=0.\ }$[34]這個定義中的行列式可以展開成一個關於${\displaystyle \lambda }$n多項式，叫做矩陣A特徵多項式，記為${\displaystyle p_{\mathbf {A} }}$。特徵多項式是一個首一多項式（最高次項係數是1的多項式）。它的根就是矩陣A特徵值[35]哈密爾頓－凱萊定理說明，如果用矩陣A本身代替多項式中的不定元${\displaystyle \lambda }$，那麼多項式的值是零矩陣[36]
${\displaystyle p_{\mathbf {A} }(\mathbf {A} )=0}$

### 正定性

 矩陣表達式 ${\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&-{\frac {1}{4}}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}}$ 正定性 不定矩陣 正定矩陣 對應二次型 ${\displaystyle Q(x,y)={\frac {1}{4}}(x^{2}-y^{2})}$ ${\displaystyle Q(x,y)={\frac {1}{4}}(x^{2}+y^{2})}$ 取值圖像 說明 正定矩陣對應的二次型的取值範圍永遠是正的， 不定矩陣對應的二次型取值則可正可負

n×n的實對稱矩陣A如果滿足對所有非零向量x ∈ Rn，對應的二次型

Q(x) ${\displaystyle =}$ xTAx

## 矩陣的計算

A−1 ${\displaystyle =}$ Adj(A) / det(A)

### 矩陣分解

LU分解將矩陣分解為一個下三角矩陣L和一個上三角矩陣U的乘積[45]。分解後的矩陣可以方便某些問題的解決。例如解線性方程組時，如果將係數矩陣A分解成A ${\displaystyle =}$ LU的形式，那麼方程式的求解可以分解為求解Ly ${\displaystyle =}$ bUx ${\displaystyle =}$ y兩步，而後兩個方程式可以十分簡潔地求解（詳見三角矩陣中「向前與向後替換」一節）。又例如在求矩陣的行列式時，如果直接計算一個矩陣A的行列式，需要計算大約(n + 1)!次加法和乘法；而如果先對矩陣做LU分解，再求行列式，就只需要大約n3次加法和乘法，大大降低了計算次數。這是因為做LU分解的複雜度大約是n3次，而後注意到LU是三角矩陣，所以求它們的行列式只需要將主對角線上元素相乘即可。

An ${\displaystyle =}$ (PDP−1)n ${\displaystyle =}$ PDP−1PDP−1...PDP−1 ${\displaystyle =}$ PDn P−1

## 矩陣的推廣

### 一般域和環上的矩陣

${\displaystyle p_{X_{\alpha }}=\left(\operatorname {min} _{\mathbf {K} }(\alpha )\right)^{r}\,}$。其中的${\displaystyle r}$是擴域L/K ${\displaystyle (\alpha )}$的階數[52]

R交換環，則${\displaystyle {\mathcal {M}}(m,\mathbf {R} )}$是一個帶單位元素R-代數，滿足結合律，但不滿足交換律。其中的矩陣仍然可以用萊布尼茲公式定義行列式。一個矩陣可逆若且唯若其行列式為環R中的可逆元素（域上的矩陣可逆只需行列式不等於0）[55]

### 矩陣群

MTM = I

(Mv) · (Mw) = v · w.[61]

### 分塊矩陣

${\displaystyle P={\begin{bmatrix}1&2&3&2\\1&2&7&5\\4&9&2&6\\6&1&5&8\end{bmatrix}}}$

${\displaystyle P_{11}={\begin{bmatrix}1&2\\1&2\end{bmatrix}},P_{12}={\begin{bmatrix}3&2\\7&5\end{bmatrix}},P_{21}={\begin{bmatrix}4&9\\6&1\end{bmatrix}},P_{22}={\begin{bmatrix}2&6\\5&8\end{bmatrix}}}$
${\displaystyle P={\begin{bmatrix}P_{11}&P_{12}\\P_{21}&P_{22}\end{bmatrix}}}$。將矩陣分塊可以使得矩陣結構清晰，在某些時候可以方便運算、證明。兩個大小相同、分塊方式也相同的矩陣可以相加。行和列的塊數符合矩陣乘法要求時，分塊矩陣也可以相乘。將矩陣分塊相乘的結果與直接相乘是一樣的。用分塊矩陣求逆，可以將高階矩陣的求逆轉化為多次低階矩陣的求逆[68]

## 應用

${\displaystyle a+ib\leftrightarrow {\begin{bmatrix}a&-b\\b&a\end{bmatrix}},}$

### 數學分析

${\displaystyle H(f)(x)=\left[{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x)\right]}$
n=2時，海森矩陣${\displaystyle {\begin{bmatrix}2&0\\0&-2\end{bmatrix}}}$的特徵值一正一負，說明函數f(x,y) = x2 − y2在 (x = 0, y = 0)處有一個鞍點（紅色點）

${\displaystyle f(x+h)=f(x)+\nabla f(x)\cdot h+{\frac {1}{2}}h^{T}H(f)(x)h+\circ \left(\|x\|^{3}\right)}$

${\displaystyle J_{f}(x)=\left[{\frac {\partial f_{i}}{\partial x_{j}}}(x)\right]_{1\leq i\leq m,1\leq j\leq n}}$。如果n>m，而${\displaystyle J_{f}(x)}$又是滿秩矩陣（秩等於m）的話，根據反函數定理，可以找到函數fx附近的一個局部的反函數[79]

${\displaystyle (\mathbf {E} )\qquad \qquad \sum _{1\leqslant i,j\leqslant n}a_{ij}{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}+\sum _{i=1}^{n}b_{i}{\frac {\partial f}{\partial x_{i}}}+cf=g.\qquad }$ 並假設${\displaystyle a_{ij}=a_{ji},}$

### 機率論與統計

${\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{i1}+\beta _{2}X_{i2}+\ldots +\beta _{p}X_{ip}+\varepsilon _{i},\qquad i=1,\ldots ,n}$

### 量子態的線性組合

1925年海森堡提出第一個量子力學模型時，使用了無限維矩陣來表示理論中作用在量子態上的算子[91]。這種做法在矩陣力學中也能見到。例如密度矩陣就是用來刻畫量子系統中「純」量子態的線性組合表示的「混合」量子態[92]

## 注釋與參考

### 腳註

1. 董可榮 2007, 第3節
2. ^ Shen, Crossley & Lun 1999
3. ^ 克萊因 2002, 第33章第4節
4. ^ Hawkins 1975
5. ^ 克萊因 2002, 第33章第4節
6. ^ The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, Paper 37, p. 247
7. ^ 克萊因 2002, 第33章第4節
8. ^ 克萊因 2002, 第33章第4節
9. ^ Cayley 1889, vol. II, p. 475–496
10. ^ Dieudonné, ed. 1978, Vol. 1, Ch. III, p. 96
11. ^ 克萊因 2002, 第33章第4節
12. ^ 克萊因 2002, 第33章第4節
13. ^ 周建華. 《矩陣》. 台灣: 中央圖書出版社. 2002. ISBN 9789576374913 （中文）.
14. ^ Brown 1991, Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)
15. ^ Brown 1991, Theorem I.2.6
16. ^ Brown 1991, Definition I.2.20
17. ^ 林志興 & 楊忠鵬 2010
18. ^ Horn & Johnson 1985, Ch. 4 and 5
19. ^ Brown 1991, I.2.21 and 22
20. ^ Greub 1975, Section III.2
21. ^ Brown 1991, Definition II.3.3
22. ^ Greub 1975, Section III.1
23. ^ Brown 1991, Theorem II.3.22
24. ^ Brown 1991, Definition I.5.13
25. ^ Brown 1991, Definition I.2.28
26. ^ 這個結論容易從矩陣乘法的定義獲得：
${\displaystyle \scriptstyle \operatorname {tr} ({\mathsf {AB}})=\sum _{i=1}^{m}\sum _{j=1}^{n}A_{ij}B_{ji}=\operatorname {tr} ({\mathsf {BA}})}$
27. ^ Brown 1991, Definition III.2.1
28. ^ Mirsky 1990, Theorem 1.4.1
29. ^ Brown 1991, Theorem III.2.12
30. ^ Brown 1991, Corollary III.2.16
31. ^ Brown 1991, Theorem III.3.18
32. ^ Brown 1991, Definition III.4.1
33. ^ Steven A. Leduc [[#CITEREFSteven A. Leduc|]], 第293頁
34. ^ Brown 1991, Definition III.4.9
35. ^ Brown 1991, Corollary III.4.10
36. ^ 王萼芳 1997, 4.2，定理3，第247頁
37. ^ Horn & Johnson 1985, Theorem 2.5.6
38. ^ Horn & Johnson 1985, Chapter 7
39. ^ Horn & Johnson 1985, Theorem 7.2.1
40. ^ Bau III & Trefethen 1997
41. ^ Householder 1975, Ch. 7
42. ^ Golub & Van Loan 1996, Algorithm 1.3.1
43. ^ Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2
44. ^ Golub & Van Loan 1996, Chapter 2.3
45. ^ Press, Flannery & Teukolsky 1992
46. ^ Stoer & Bulirsch 2002, Section 4.1
47. ^ Horn & Johnson 1985, Theorem 2.5.4
48. ^ Horn & Johnson 1985, Ch. 3.1, 3.2
49. ^ Arnold & Cooke 1992, Sections 14.5, 7, 8
50. ^ Bronson 1989, Ch. 15
51. ^ Coburn 1955, Ch. V
52. ^ Ash 2012, Chapter II
53. ^ Lang 2002, Chapter XIII
54. ^ Lang 2002, XVII.1, p. 643
55. ^ Lang 2002, Proposition XIII.4.16
56. ^ Greub 1975, Section III.3
57. ^ Greub 1975, Section III.3.13
58. ^ Baker 2003, Def. 1.30
59. ^ Baker 2003, Theorem 1.2
60. ^ Artin 1991, Chapter 4.5
61. ^ Artin 1991, Theorem 4.5.13
62. ^ Rowen 2008, Example 19.2, p. 198
63. ^ Itõ, ed. 1987
64. ^ Thankappan 1993
65. ^ Thankappan 1993
66. ^ Thankappan 1993
67. ^ Faliva & Zoia 2008
68. ^ 居余馬 2002, 2.6
69. ^ Fudenberg & Tirole 1983, Section 1.1.1
70. ^ Manning 1999, Section 15.3.4
71. ^ Ward 1997, Ch. 2.8
72. ^ Stinson 2005, Ch. 1.1.5 and 1.2.4
73. ^ Association for Computing Machinery 1979, Ch. 7
74. ^ Godsil & Royle 2004, Ch. 8.1
75. ^ Punnen 2002
76. ^ Lang 1987a, Ch. XVI.6
77. ^ Nocedal 2006, Ch. 16
78. ^ Lang 1987a, Ch. XVI.1
79. ^ Lang 1987a, Ch. XVI.5
80. ^ Gilbarg & Trudinger 2001
81. ^ Šolin 2005, Ch. 2.5
82. ^ 伊澤爾萊斯 2005, Ch. 8
83. ^ Latouche & Ramaswami 1999
84. ^ Mehata & Srinivasan 1978, Ch. 2.8
85. ^ Krzanowski 1988, Ch. 2.2., p. 60
86. ^ Krzanowski 1988, Ch. 4.1
87. ^ Conrey 2007
88. ^ Zabrodin, Brezin & Kazakov et al. 2006
89. ^ Itzykson & Zuber 1980, Ch. 2
90. ^ Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi－Maskawa matrix)
91. ^ Schiff 1968, Ch. 6
92. ^ Bohm 2001, sections II.4 and II.8
93. ^ Weinberg 1995, Ch. 3
94. ^ Wherrett 1987, part II
95. ^ Riley, Hobson & Bence 1997, 7.17
96. ^ Guenther 1990, Ch. 5

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