GCT矩阵.pdf

线性代数
1 矩阵的概念
引例某地区有甲、乙、丙3个企业, 生产A、B、
C、D 4 种产品,每种产品的单价与单位利润如下
二、矩阵单价单位利润
b b ⎛ b11 b12 ⎞
A 11 12 ⎜⎟
b b
B b21 b22 ⎜ 21 22 ⎟
⇒⎜⎟
b31 b32
C b31 b32 ⎜⎟
⎜ b b ⎟
D b41 b42 ⎝ 41 42 ⎠
101 102
(1)定义
个数
⎧ a11 x1 + a12 x2 +"+ a1n xn = b1 ¾ m × n
⎪ a , i = 1,2,",m; j = 1,2,",n
⎪ a21 x1 + a22 x2 +"+ a2n xn = b2 ij
⎨
⎪""""""""""" 排成 m 行 n 列的长方形数表, 用圆括号或方括号
⎪括起来, 称为矩阵. 记为A, B, C , "
⎩am1 x1 + am2 x2 + "+ amn xn = bm
如
⎛ a11 a12 " a1n ⎞
⎛ a a " a ⎞⎛⎞aa" a b ⎜⎟
⎜ 11 12 1n ⎟ 11 12 1n 1 a a " a
⎜⎟⎜ 21 22 2n ⎟= a
⎜ a21 a22 " a2n ⎟ aa21 22" a 2n b 2 A = ( ij )m×n
⇒,⎜⎟⎜" " " " ⎟
⎜" " " " ⎟⎜⎟""""" ⎜⎟
⎜" ⎟
⎜⎟⎜⎟⎝ am1 am2 amn ⎠
⎝ am1 am2 " amn ⎠⎝⎠aamm12" a mnm b
aij —矩阵的元素
系数矩阵增广矩阵
103 104
(2)几种特殊类型的矩阵:
¾ 定义同类型矩阵与两个矩阵的相等
‰ 行(列)矩阵
A = a ,B = b
设( ij )m×n ( ij )s×t
只有一行(列)的矩阵
若 m =s , n = t , 则称 A,B 为同类型矩阵
⎛⎞b1
若还有⎜⎟
(a1 a2 " an ) b
⎜⎟2
aij = bij , i = 1,2,",m;
()a1 , a2 , ", an ⎜⎟#
j = 1,2,",n ⎜⎟
⎝⎠bn
则称 A,B相等,记为 A = B
105 106
18
线性代数
‰ 零矩阵特殊类型的方阵:
⎛ 0 0 " 0 ⎞
⎜⎟
⎜ 0 0 " 0 ⎟¾ 对角矩阵
O = = 0
m ×n ⎜" " " " ⎟
⎜⎟ a 0 " 0
⎜ 0 0 " 0 ⎟⎛ 11 ⎞
⎝⎠ m×n ⎜⎟⎛⎞
⎜ 0 a22 " 0 ⎟⎜ 0 ⎟
‰ 方阵⎜⎟⎜⎟
" " " " ⎜ 0 ⎟
⎜⎟⎝⎠
⎛ a11 a12 " a1n ⎞方阵的主对角线⎝ 0 0 " ann ⎠n×n
⎜⎟主对角元
⎜ a 21 a 22 " a 2 n ⎟
a = 0, i ≠ j,i, j = 1,2,",n
⎜" " " " ⎟方阵的行列式 ij
⎜⎟
a a " a A = a 也记为 diag(a11 ,a22 ,",ann )
⎝ n1 n 2 nn ⎠ n×n ij n
detAa= det(ij ) n× n
107 108
¾数量矩阵¾ 三角矩阵
⎛ a 0 " 0 ⎞⎛ a a " a ⎞
⎜⎟⎜ 11 12 1n ⎟⎛⎞
0 a " 0 ⎜⎟
⎜⎟⎜ 0 a22 " a2n ⎟
⎜⎟
⎜" " " "⎟⎜" " " " ⎟ 0
⎜⎟⎜⎟⎜⎟
⎜ 0 0 " a ⎟⎜⎟⎝⎠
⎝⎠n×n ⎝ 0 0 " ann ⎠
上三角
¾单位矩阵
⎛ a 0 " 0 ⎞
⎜ 11 ⎟⎛⎞
⎜ 0 ⎟
⎛ 1 0 " 0 ⎞⎜ a21 a22 " 0 ⎟
⎜⎟⎜⎟
0 1 " 0 ⎜" " " " ⎟
⎜⎟记为 E 或 I ⎜⎟⎜⎟
⎜⎟⎜⎟⎝⎠
" " " " ⎝ an1 an2 " ann ⎠
⎜⎟
⎜⎟下三角
⎝ 0 0 " 1 ⎠n×n
109 110
¾对称矩阵与反对称矩阵
2 矩阵的运算
⎛ a11 a12 " a1n ⎞⎛ 1 2 − 5⎞
⎜⎟⎜⎟注意:
a a " a aij = a ji ,
⎜ 21 22 2n ⎟⎜ 2 0 7 ⎟
⎜⎟ i, j = 1,2,",n ⎜⎟什么样的矩阵之间能进行这种运算?
" " " " ⎝− 5 7 3 ⎠‰
⎜⎟