矩阵本文重定向自 矩阵

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

m-by-n matrix”的各地常用别名

“横排（row）”的各地常用别名

“纵排（column）”的各地常用别名

${\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}&\dots &a_{1j}&\dots &a_{1n}\\a_{21}&a_{22}&a_{23}&\dots &a_{2j}&\dots &a_{2n}\\a_{31}&a_{32}&a_{33}&\dots &a_{3j}&\dots &a_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots \\a_{i1}&a_{i2}&a_{i3}&\dots &a_{ij}&\dots &a_{in}\\\vdots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots \\a_{m1}&a_{n2}&a_{m3}&\dots &a_{mi}&\dots &a_{mn}\end{bmatrix}}}$

定义

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

标记

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

矩阵的基本运算

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

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

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

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

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

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

矩阵乘法

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

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

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

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

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

线性方程组

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

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

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

线性变换

 推移，幅度m=1.25. 水平镜射变换 “挤压”变换，压缩程度r=3/2 伸缩，3/2倍 旋转，左转30° ${\displaystyle {\begin{bmatrix}1&1.25\\0&1\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}-1&0\\0&1\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {2}{3}}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}{\frac {3}{2}}&0\\0&{\frac {3}{2}}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}\cos({\frac {\pi }{6}})&-\sin({\frac {\pi }{6}})\\\sin({\frac {\pi }{6}})&\cos({\frac {\pi }{6}})\end{bmatrix}}}$

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

方块矩阵

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

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

行列式

R2里的一个线性变换f将蓝色图形变成绿色图形，面积不变，而顺时针排布的向量x1和x2的变成了逆时针排布。对应的矩阵行列式是-1.

2×2矩阵的行列式是

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

3×3矩阵的行列式由6项组成。更高维矩阵的行列式则可以使用莱布尼兹公式写出，或使用拉普拉斯展开由低一维的矩阵行列式递推得出。

特征值与特征向量

${\displaystyle n\times n}$的方块矩阵${\displaystyle \mathbf {A} }$的一个特征值和对应特征向量是满足

${\displaystyle \mathbf {Av} =\lambda \mathbf {v} }$的标量${\displaystyle \lambda }$以及非零向量${\displaystyle \mathbf {v} }$。特征值和特征向量的概念对研究线性变换很有帮助。一个线性变换可以通过它对应的矩阵在向量上的作用来可视化。一般来说，一个向量在经过映射之后可以变为任何可能的向量，而特征向量具有更好的性质。假设在给定的基底下，一个线性变换对应着某个矩阵${\displaystyle \mathbf {A} }$，如果一个向量${\displaystyle \mathbf {x} }$可以写成矩阵的几个特征向量的线性组合：
${\displaystyle \mathbf {x} =c_{1}\mathbf {x} _{\lambda _{1}}+c_{2}\mathbf {x} _{\lambda _{2}}+\cdots +c_{k}\mathbf {x} _{\lambda _{k}}}$

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

${\displaystyle \det(\lambda \mathbf {I} _{n}-\mathbf {A} )=0.\ }$这个定义中的行列式可以展开成一个关于${\displaystyle \lambda }$n多项式，叫做矩阵A特征多项式，记为${\displaystyle p_{\mathbf {A} }}$。特征多项式是一个首一多项式（最高次项系数是1的多项式）。它的根就是矩阵${\displaystyle \mathbf {A} }$特征值。哈密尔顿－凯莱定理说明，如果用矩阵${\displaystyle \mathbf {A} }$本身代替多项式中的不定元${\displaystyle \lambda }$，那么多项式的值是零矩阵：
${\displaystyle p_{\mathbf {A} }(\mathbf {A} )=0}$

正定性

 矩阵表达式 ${\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&-{\frac {1}{4}}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}{\frac {1}{4}}&0\\0&{\frac {1}{4}}\end{bmatrix}}}$ 正定性 不定矩阵 正定矩阵 对应二次型 ${\displaystyle Q(x,y)={\frac {1}{4}}(x^{2}-y^{2})}$ ${\displaystyle Q(x,y)={\frac {1}{4}}(x^{2}+y^{2})}$ 取值图像 说明 正定矩阵对应的二次型的取值范围永远是正的，不定矩阵对应的二次型取值则可正可负

${\displaystyle n\times n}$的实对称矩阵${\displaystyle \mathbf {A} }$如果满足对所有非零向量${\displaystyle \mathbf {x} \in \mathbf {R} ^{n}}$，对应的二次型

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

矩阵的计算

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

矩阵分解

LU分解将矩阵分解为一个下三角矩阵${\displaystyle \mathbf {L} }$和一个上三角矩阵${\displaystyle \mathbf {U} }$的乘积。分解后的矩阵可以方便某些问题的解决。例如解线性方程组时，如果将系数矩阵${\displaystyle \mathbf {A} }$分解成${\displaystyle \mathbf {A} =\mathbf {LU} }$的形式，那么方程的求解可以分解为求解${\displaystyle \mathbf {Ly} =\mathbf {b} }$${\displaystyle \mathbf {Ux} =\mathbf {y} }$两步，而后两个方程可以十分简洁地求解（详见三角矩阵中“向前与向后替换”一节）。又例如在求矩阵的行列式时，如果直接计算一个矩阵${\displaystyle \mathbf {A} }$的行列式，需要计算大约${\displaystyle (n+1)!}$次加法和乘法；而如果先对矩阵做${\displaystyle \mathbf {LU} }$分解，再求行列式，就只需要大约${\displaystyle n^{3}}$次加法和乘法，大大降低了计算次数。这是因为做${\displaystyle \mathbf {LU} }$分解的复杂度大约是${\displaystyle n^{3}}$次，而后注意到${\displaystyle \mathbf {L} }$${\displaystyle \mathbf {U} }$是三角矩阵，所以求它们的行列式只需要将主对角线上元素相乘即可。

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

矩阵的推广

一般域和环上的矩阵

${\displaystyle p_{X_{\alpha }}=\left(\operatorname {min} _{\mathbf {K} }(\alpha )\right)^{r}\,}$。其中的${\displaystyle r}$是扩域${\displaystyle \mathbf {L/K} }$ ${\displaystyle (\alpha )}$的阶数。

${\displaystyle \mathbf {R} }$交换环，则${\displaystyle {\mathcal {M}}(m,\mathbf {R} )}$是一个带单位元${\displaystyle \mathbf {R} }$-代数，满足结合律，但不满足交换律。其中的矩阵仍然可以用莱布尼兹公式定义行列式。一个矩阵可逆当且仅当其行列式为环${\displaystyle \mathbf {R} }$中的可逆元（域上的矩阵可逆只需行列式不等于0）。

矩阵群

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

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

分块矩阵

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

${\displaystyle P_{11}={\begin{bmatrix}1&2\\1&2\end{bmatrix}},P_{12}={\begin{bmatrix}3&2\\7&5\end{bmatrix}},P_{21}={\begin{bmatrix}4&9\\6&1\end{bmatrix}},P_{22}={\begin{bmatrix}2&6\\5&8\end{bmatrix}}}$
${\displaystyle P={\begin{bmatrix}P_{11}&P_{12}\\P_{21}&P_{22}\end{bmatrix}}}$。将矩阵分块可以使得矩阵结构清晰，在某些时候可以方便运算、证明。两个大小相同、分块方式也相同的矩阵可以相加。行和列的块数符合矩阵乘法要求时，分块矩阵也可以相乘。将矩阵分块相乘的结果与直接相乘是一样的。用分块矩阵求逆，可以将高阶矩阵的求逆转化为多次低阶矩阵的求逆。

应用

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

数学分析

${\displaystyle H(f)(x)=\left[{\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x)\right]}$
${\displaystyle n=2}$时，海森矩阵${\displaystyle {\begin{bmatrix}2&0\\0&-2\end{bmatrix}}}$的特征值一正一负，说明函数${\displaystyle f(x,y)=x^{2}-y^{2}}$${\displaystyle (x=0,y=0)}$处有一个鞍点（红色点）

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

${\displaystyle J_{f}(x)=\left[{\frac {\partial f_{i}}{\partial x_{j}}}(x)\right]_{1\leq i\leq m,1\leq j\leq n}}$。如果${\displaystyle n>m}$，而${\displaystyle J_{f}(x)}$又是满秩矩阵（秩等于${\displaystyle m}$）的话，根据反函数定理，可以找到函数${\displaystyle f}$x附近的一个局部的反函数。

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

概率论与统计

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

量子态的线性组合

1925年海森堡提出第一个量子力学模型时，使用了无限维矩阵来表示理论中作用在量子态上的算子。这种做法在矩阵力学中也能见到。例如密度矩阵就是用来刻画量子系统中“纯”量子态的线性组合表示的“混合”量子态。

注释与参考

脚注

1. 董可荣 2007, 第3节
2. ^ Shen, Crossley & Lun 1999
3. ^ 克莱因 2002, 第33章第4节
4. ^ Hawkins 1975
5. ^ 克莱因 2002, 第33章第4节
6. ^ The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, Paper 37, p. 247
7. ^ 克莱因 2002, 第33章第4节
8. ^ 克莱因 2002, 第33章第4节
9. ^ Cayley 1889, vol. II, p. 475–496
10. ^ Dieudonné, ed. 1978, Vol. 1, Ch. III, p. 96
11. ^ 克莱因 2002, 第33章第4节
12. ^ 克莱因 2002, 第33章第4节
13. ^ 周建华. 《矩陣》. 台湾: 中央图书出版社. 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. ^ 林志兴 & 杨忠鹏 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. ^ 这个结论容易从矩阵乘法的定义获得：
${\displaystyle \scriptstyle \operatorname {tr} ({\mathsf {AB}})=\sum _{i=1}^{m}\sum _{j=1}^{n}A_{ij}B_{ji}=\operatorname {tr} ({\mathsf {BA}})}$
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页
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页
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. ^ 居余马 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. ^ 伊泽尔莱斯 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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