# 矩陣

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

m-by-n matrix」的各地常用別名

「橫排（row）」的各地常用別名

「縱排（column）」的各地常用別名

${\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}&\dots &a_{1j}&\dots &a_{1n}\\a_{21}&a_{22}&a_{23}&\dots &a_{2j}&\dots &a_{2n}\\a_{31}&a_{32}&a_{33}&\dots &a_{3j}&\dots &a_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots \\a_{i1}&a_{i2}&a_{i3}&\dots &a_{ij}&\dots &a_{in}\\\vdots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&a_{m3}&\dots &a_{mj}&\dots &a_{mn}\end{bmatrix}}}$

## 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

${\displaystyle (\mathbf {A} \pm \mathbf {B} )_{i,j}=\mathbf {A} _{i,j}\pm \mathbf {B} _{i,j}}$

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

${\displaystyle (c\mathbf {A} )_{i,j}=c\cdot \mathbf {A} _{i,j}}$
${\displaystyle 2\cdot {\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot (-3)\\2\cdot 4&2\cdot (-2)&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}}$

${\displaystyle (\mathbf {A} ^{\mathrm {T} })_{i,j}=\mathbf {A} _{j,i}}$
${\displaystyle {\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{T}={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}}$

${\displaystyle (\mathbf {A} +\mathbf {B} )^{\mathrm {T} }=\mathbf {A} ^{\mathrm {T} }+\mathbf {B} ^{\mathrm {T} }}$
${\displaystyle c(\mathbf {A} +\mathbf {B} )=c\mathbf {A} +c\mathbf {B} }$

${\displaystyle c(\mathbf {A} ^{\mathrm {T} })=c(\mathbf {A} )^{\mathrm {T} }}$

## 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：${\displaystyle (\mathbf {AB} )\mathbf {C} =\mathbf {A} (\mathbf {BC} )}$
• 左分配律：${\displaystyle (\mathbf {A} +\mathbf {B} )\mathbf {C} =\mathbf {AC} +\mathbf {BC} }$
• 右分配律：${\displaystyle \mathbf {C} (\mathbf {A} +\mathbf {B} )=\mathbf {CA} +\mathbf {CB} }$

${\displaystyle c(\mathbf {AB} )=(c\mathbf {A} )\mathbf {B} =\mathbf {A} (c\mathbf {B} )}$
${\displaystyle (\mathbf {AB} )^{\mathrm {T} }=\mathbf {B} ^{\mathrm {T} }\mathbf {A} ^{\mathrm {T} }}$

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

### 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° ${\displaystyle {\begin{bmatrix}1&1.25\\0&1\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}-1&0\\0&1\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {2}{3}}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {3}{2}}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}\cos({\frac {\pi }{6}})&-\sin({\frac {\pi }{6}})\\\sin({\frac {\pi }{6}})&\cos({\frac {\pi }{6}})\end{bmatrix}}}$

${\displaystyle (g\circ f)(x)=g(f(x))=g(\mathbf {Ax} )=\mathbf {B} (\mathbf {Ax} )=(\mathbf {BA} )\mathbf {x} }$

## 方塊矩陣

${\displaystyle \mathbf {AB} =\mathbf {I} _{n}}$

${\displaystyle {\begin{bmatrix}d_{11}&0&0\\0&d_{22}&0\\0&0&d_{33}\\\end{bmatrix}}}$（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裡的一個線性變換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

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

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[22]，或使用拉普拉斯展開由低一維的矩陣行列式遞推得出[23]

### 特徵值與特徵向量

${\displaystyle n\times n}$的方塊矩陣${\displaystyle \mathbf {A} }$的一個特徵值和對應特徵向量是滿足

${\displaystyle \mathbf {Av} =\lambda \mathbf {v} }$[27]的標量${\displaystyle \lambda }$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[28]。假設在給定的基底下，一個線性變換對應着某個矩陣${\displaystyle \mathbf {A} }$，如果一個向量${\displaystyle \mathbf {x} }$可以寫成矩陣的幾個特徵向量的線性組合：
${\displaystyle \mathbf {x} =c_{1}\mathbf {x} _{\lambda _{1}}+c_{2}\mathbf {x} _{\lambda _{2}}+\cdots +c_{k}\mathbf {x} _{\lambda _{k}}}$

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

${\displaystyle \det(\lambda \mathbf {I} _{n}-\mathbf {A} )=0.\ }$[29]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${\displaystyle \mathbf {A} }$特徵值[30]哈密爾頓－凱萊定理說明，如果用矩陣${\displaystyle \mathbf {A} }$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[31]
${\displaystyle p_{\mathbf {A} }(\mathbf {A} )=0}$

### 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

${\displaystyle n\times n}$的實對稱矩陣${\displaystyle \mathbf {A} }$如果滿足對所有非零向量${\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}$，對應的二次型

${\displaystyle Q(\mathbf {x} )=\mathbf {x} ^{\mathrm {T} }\mathbf {Ax} }$

## 矩陣的計算

${\displaystyle \mathbf {A} ^{-1}={\frac {\operatorname {adj} (\mathbf {A} )}{\det(\mathbf {A} )}}}$

### 矩陣分解

LU分解將矩陣分解為一個下三角矩陣${\displaystyle \mathbf {L} }$和一個上三角矩陣${\displaystyle \mathbf {U} }$的乘積[40]。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${\displaystyle \mathbf {A} }$分解成${\displaystyle \mathbf {A} =\mathbf {LU} }$的形式，那麼方程的求解可以分解為求解${\displaystyle \mathbf {Ly} =\mathbf {b} }$${\displaystyle \mathbf {Ux} =\mathbf {y} }$兩步，而後兩個方程可以十分簡潔地求解（詳見三角矩陣中「向前與向後替換」一節）。又例如在求矩陣的行列式時，如果直接計算一個矩陣${\displaystyle \mathbf {A} }$的行列式，需要計算大約${\displaystyle (n+1)!}$次加法和乘法；而如果先對矩陣做${\displaystyle \mathbf {LU} }$分解，再求行列式，就只需要大約${\displaystyle n^{3}}$次加法和乘法，大大降低了計算次數。這是因為做${\displaystyle \mathbf {LU} }$分解的複雜度大約是${\displaystyle n^{3}}$次，而後注意到${\displaystyle \mathbf {L} }$${\displaystyle \mathbf {U} }$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。

${\displaystyle \mathbf {A} ^{n}=(\mathbf {PDP} ^{-1})^{n}=\mathbf {PDP} ^{-1}\mathbf {PDP} ^{-1}\ldots \mathbf {PDP} ^{-1}=\mathbf {PD} ^{n}\mathbf {P} ^{-1}}$

## 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${\displaystyle \mathbf {L/K} }$ ${\displaystyle (\alpha )}$的階數[47]

${\displaystyle \mathbf {R} }$交換環，則${\displaystyle {\mathcal {M}}(m,\mathbf {R} )}$是一個帶單位元${\displaystyle \mathbf {R} }$-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。一個矩陣可逆當且僅當其行列式為環${\displaystyle \mathbf {R} }$中的可逆元（域上的矩陣可逆只需行列式不等於0）[50]

### 矩陣群

${\displaystyle \mathbf {M} ^{\mathrm {T} }\mathbf {M} =\mathbf {I} }$

${\displaystyle (\mathbf {Mv} )\cdot (\mathbf {Mw} )=\mathbf {v} \cdot \mathbf {w} }$[56]

### 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[65]

## 應用

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

### 數學分析

${\displaystyle H(f)(x)=\left[{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x)\right]}$
${\displaystyle n=2}$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${\displaystyle f(x,y)=x^{2}-y^{2}}$${\displaystyle (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${\displaystyle n>m}$，而${\displaystyle J_{f}(x)}$又是滿秩矩陣（秩等於${\displaystyle m}$）的話，根據反函數定理，可以找到函數${\displaystyle f}$x附近的一個局部的反函數[76]

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

### 概率論與統計

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

### 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[88]。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[89]

## 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. ^ The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, Paper 37頁面存檔備份，存於網際網路檔案館）, p. 247
6. ^ Cayley 1889, vol. II, p. 475–496
7. ^ Dieudonné, ed. 1978, Vol. 1, Ch. III, p. 96
8. ^ 周建華. 《矩陣》. 台灣: 中央圖書出版社. 2002. ISBN 9789576374913 （中文）.
9. ^ Brown 1991, Definition I.2.1 (addition), Definition I.2.4 (scalar multiplication), and Definition I.2.33 (transpose)
10. ^ Brown 1991, Theorem I.2.6
11. ^ Brown 1991, Definition I.2.20
12. ^ 林志興 & 楊忠鵬 2010
13. ^ Horn & Johnson 1985, Ch. 4 and 5
14. ^ Brown 1991, I.2.21 and 22
15. ^ Greub 1975, Section III.2
16. ^ Brown 1991, Definition II.3.3
17. ^ Greub 1975, Section III.1
18. ^ Brown 1991, Theorem II.3.22
19. ^ Brown 1991, Definition I.5.13
20. ^ Brown 1991, Definition I.2.28
21. ^ 這個結論容易從矩陣乘法的定義獲得：
${\displaystyle \scriptstyle \operatorname {tr} ({\mathsf {AB}})=\sum _{i=1}^{m}\sum _{j=1}^{n}A_{ij}B_{ji}=\operatorname {tr} ({\mathsf {BA}})}$
22. ^ Brown 1991, Definition III.2.1
23. ^ Mirsky 1990, Theorem 1.4.1
24. ^ Brown 1991, Theorem III.2.12
25. ^ Brown 1991, Corollary III.2.16
26. ^ Brown 1991, Theorem III.3.18
27. ^ Brown 1991, Definition III.4.1
28. ^ Steven A. Leduc [[#CITEREFSteven A. Leduc|]], 第293頁
29. ^ Brown 1991, Definition III.4.9
30. ^ Brown 1991, Corollary III.4.10
31. ^ 王萼芳 1997, 4.2，定理3，第247頁
32. ^ Horn & Johnson 1985, Theorem 2.5.6
33. ^ Horn & Johnson 1985, Chapter 7
34. ^ Horn & Johnson 1985, Theorem 7.2.1
35. ^ Bau III & Trefethen 1997
36. ^ Householder 1975, Ch. 7
37. ^ Golub & Van Loan 1996, Algorithm 1.3.1
38. ^ Golub & Van Loan 1996, Chapters 9 and 10, esp. section 10.2
39. ^ Golub & Van Loan 1996, Chapter 2.3
40. ^ Press, Flannery & Teukolsky 1992
41. ^ Stoer & Bulirsch 2002, Section 4.1
42. ^ Horn & Johnson 1985, Theorem 2.5.4
43. ^ Horn & Johnson 1985, Ch. 3.1, 3.2
44. ^ Arnold & Cooke 1992, Sections 14.5, 7, 8
45. ^ Bronson 1989, Ch. 15
46. ^ Coburn 1955, Ch. V
47. ^ Ash 2012, Chapter II
48. ^ Lang 2002, Chapter XIII
49. ^ Lang 2002, XVII.1, p. 643
50. ^ Lang 2002, Proposition XIII.4.16
51. ^ Greub 1975, Section III.3
52. ^ Greub 1975, Section III.3.13
53. ^ Baker 2003, Def. 1.30
54. ^ Baker 2003, Theorem 1.2
55. ^ Artin 1991, Chapter 4.5
56. ^ Artin 1991, Theorem 4.5.13
57. ^ Rowen 2008, Example 19.2, p. 198
58. ^ Itõ, ed. 1987
59. ^ Thankappan 1993
60. ^ Thankappan 1993
61. ^ Thankappan 1993
62. ^ "Empty Matrix: A matrix is empty if either its row or column dimension is zero". O-Matrix v6 User Guide. （原始內容存檔於2009-04-29）.
63. ^ Matrix - MATLAB Data Structures. system.nada.kth.se. （原始內容存檔於2009-12-28）. A matrix having at least one dimension equal to zero is called an empty matrix
64. ^ Faliva & Zoia 2008
65. ^ 居余馬 2002, 2.6
66. ^ Fudenberg & Tirole 1983, Section 1.1.1
67. ^ Manning 1999, Section 15.3.4
68. ^ Ward 1997, Ch. 2.8
69. ^ Stinson 2005, Ch. 1.1.5 and 1.2.4
70. ^ Association for Computing Machinery 1979, Ch. 7
71. ^ Godsil & Royle 2004, Ch. 8.1
72. ^ Punnen 2002
73. ^ Lang 1987a, Ch. XVI.6
74. ^ Nocedal 2006, Ch. 16
75. ^ Lang 1987a, Ch. XVI.1
76. ^ Lang 1987a, Ch. XVI.5
77. ^ Gilbarg & Trudinger 2001
78. ^ Šolin 2005, Ch. 2.5
79. ^ 伊澤爾萊斯 2005, Ch. 8
80. ^ Latouche & Ramaswami 1999
81. ^ Mehata & Srinivasan 1978, Ch. 2.8
82. ^ Krzanowski 1988, Ch. 2.2., p. 60
83. ^ Krzanowski 1988, Ch. 4.1
84. ^ Conrey 2007
85. ^ Zabrodin, Brezin & Kazakov et al. 2006
86. ^ Itzykson & Zuber 1980, Ch. 2
87. ^ Burgess & Moore 2007, section 1.6.3. (SU(3)), section 2.4.3.2. (Kobayashi－Maskawa matrix)
88. ^ Schiff 1968, Ch. 6
89. ^ Bohm 2001, sections II.4 and II.8
90. ^ Weinberg 1995, Ch. 3
91. ^ Wherrett 1987, part II
92. ^ Riley, Hobson & Bence 1997, 7.17
93. ^ Guenther 1990, Ch. 5

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