临沂大学《线性代数》课件 矩阵及其运算

临沂大学线性代数E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410线性方程组和矩阵E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410线性方程组矩阵的定义E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410线性方程组E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410其中aij

表示第i个方程中第j

个未知量xj

的系数，

1,

2,

,

;

1,

2,

,

.12222212

a

x

a

x

a

x

a

x

am

x

am

x

1

,2

,

a

n

xn

bn

na

x

b2

2

,

amn

xn

bm设有n

个未知数m

个方程的线性方程组当

1

,

2

,

,

不全为零时，称为非齐次线性方程组.常数项i

m

j

nb

b

bm1121(1)111

1

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

am1x1

am2x2

amn

xn

0.称为齐次线性方程组.n元线性方程组简称为线性方程组或方程组.当常数项全为零,

即b1

=b2

=

=bm

=0

时,方程组成为

a11x1

a12

x2

a1nxn

0,a21x1

a22

x2

a2nxn

0,(2)E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410对于线性方程组，需要讨论如下问题：(1)方程组是否有解？(2)在方程组有解时，解是否惟一？(3)如果有多个解，如何求出它的所有解？E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410南北朝的《孙子算经》：

“鸡兔同笼”问题：“今有雉兔同笼，上有三十五头，下有九十四足，

问雉兔各几何？

”设鸡有x

只，兔有

y

只，根据题意，得：

x

y

35

(1)

2x

4y

94

(2)(2)-(1)2:2y

24

所以

y

12,x

23E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410解的情况完全由其系数

aij

(i1,

2,

,

m;

j1,

2,

,n)和常数项bi

(i

1,

2,

,

m)

决定.

a11x1

a12

x2

a21x1

a22

x2

am1x1

am2x2

a1n

xn

b1

,

a2n

xn

b2

,

amn

xn

bmE6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

aaaaaaaaab1b2b该数表就

表示方程组

a12

x2a22

x2a

xm2

2

a1n

xn

b1

,a2n

xn

b2

,a

x

bmn

n

m

a11x

21a

x1x

11

1n2n11211222am2mnm1m1mE6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵的定义E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410a1n

a2n

amn

矩阵的(i,j)元定义

由m

n

个数aij

构成的m行

n

列的矩形数表称为一个

m

n矩阵,

数aij第j列

a11

a21

am1第i行aaa1222m2E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵常用大写英文字母表示,

a11

a12

记为：A

a21

a22A

aij

m

n有时也记为如

A

、B

、C

等.a1n

am1

am2表明矩阵的阶数构成矩阵的元素

a2n

.

amn

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410根据矩阵的元素所属的数域，可以将矩阵分为：

i

2

3

2

3

1

4

实矩阵：由实数构成的矩阵.

例如：

2

3

0

1

3

2

0

i

i

复矩阵：由复数构成的矩阵.

例如：

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410为零矩阵,

记为：

0

O

0

0零矩阵通常简记为

O

.

几类特殊矩阵：1、零矩阵：如果矩阵中的每一个元素均为零,0

0

0

m

n则该矩阵称000E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B4102.

行矩阵与列矩阵行数为

1

的矩阵：

a1，

a2

，

，

am

称为行矩阵，也称为m维行向量；列数为1的矩阵:

b1

b2

bn

称为列矩阵，也称n

维列向量.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

a11

A

a21

an13.

方阵：行数和列数相等的矩阵a1n

a2n

ann

n

n为一个n

阶方阵.aaa例：1222n2E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

a11

a21

am10aa22

a11

a12

0

a22

0

0——上三角阵0

0

amm

m

m在方阵中还有几个特殊形式的矩阵

:a1n

a2n

ann

n

n——下三角阵m2E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B4100a220

a11

0

0——单位阵,记为En也是下三角矩阵——对角阵对角阵：既是上三角，0

0

ann

n

n0

0

1

n

n

1

0

0010E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵相等的概念：如果两个矩阵行数相等、列数也相等时，称这两

个矩阵是同型矩阵.如果矩阵A

aij

与B

bij

是同型矩阵，并且它们对应的元素相等，即aij

bij

i1,2,,m;j1,2,,n

则称矩阵A与B矩阵相等，记作

A

B

.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

a11

a1nxn

b1

,

a2nxn

b2

,

A=

a21

amn

xn

bm

，

am1

a11x1

a12

x2

a21x1

a22

x2

对于非齐次线性方程组方程组的系数矩阵a1n

a2n

amn

am1x1

am2x2

aaa1222m2E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

b1

未知数矩阵

b

b2

bm

a11

a21

am1

x1

x

x2

xn

A~

a1nxn

b1

,

a2nxn

b2

,

amn

xn

bm，b1

b2

bm

方程组的增广矩阵

a11x1

a12

x2

a21x1

a22

x2

常数项矩阵aaa

am1x1

am2x2

aa1n2n1222am2mnE6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

y1

a11x1

a12

x2

y2

a21x1

a22

x2

从变量x1

,

x2

，

,

xn

到变量y1

,

y2

,

,

ym

的线性变换.变量

x1

,

x2

，

,

xn

与变量y1

,

y2

,

,

ym

之间的关系式

a1nxn

,

a2nxn

,

a

x

ym

am1x1

am2x2

A

aij

m

nmn

nE6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410从变量x1

,

x2

，

,

xn

到变量y1

,

y2

,

,

yn

的恒等变换.

1

0

E

0

1

0

0

y1

x1

,y2

x2

,

yn

xn0

0

1

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵的运算E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410数与矩阵相乘矩阵与矩阵相乘

矩阵的转置方阵的行列式矩阵的加法

矩阵的线性运算E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵的加法E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410同型定义

设有两个m

n

矩阵

A

aij

和

B

bij

，那么矩阵A

与B

的和记作A

B，规定为

a11

b11

A

B

a21

b21

am1

bm1

a1n

b1n

a2n

b2n

.

amn

bmn

a12

b12a22

b22注意am2

bm2E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B4102

35

2

3

,

B

2

1+3

A

B

3

22

1

4

0

1

设

A

31

,0

2

35

0

5

5114

1例

则3

5

.

1114E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵的加法运算具有性质

:(1)交换律：

A

B

B

A.(2)结合律：A

B

C

A

B

C

.(3)对任意矩阵

A

：A

O

O

A

A.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410数与矩阵相乘E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410设矩阵A

aij

m

n，k

为一个数,定义数

k

与矩阵A

的乘法为：kA

kaij

.

1

1

2

2

例：设A

,

则

2A

.

1

1

2

2E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410数与矩阵的乘法具有如下性质：⑴对任意矩阵Am

n

，有1Am

n

Am

n

;

⑵对任意矩阵Am

n

、Bm

n

以及任意数k

、l

，有(k+l)A

kA+lA,

k(A

B)

kA

kB⑶对任意矩阵Am

n

以及任意数

k

、l，有(kl)Am

n

k(lAm

n

)

l(kAm

n

)E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410对矩阵

A

aij

m

n

，定义其负矩阵为

A

(

1)

A

aij

.

定义两个同型矩阵的减法为：A

B

A

B

aij

bij

.

1

3

5

7

9

11

例：

设

A

,

B

,

2

4

6

8

10

12

6

6

6

则

A

B

6

6

6

.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵与矩阵相乘E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B4101

、矩阵乘法的定义设矩阵

A

aij

m

n

，B

bij

n

p

，定义矩阵

C

cij

m

p

，其中

cij

ai

1b1j

ai2b2j

ainbnj

.称矩阵C

称为矩阵A

与B

的乘积,

记为C

AB.

C

A

B.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

1

1

1

例

设A

，B

，则由矩阵乘法定义

2

2

3

2

1

2

2

3

2

A

B

1

4

3

1

2

2

1

6

2

7

5

3

8

5

7

2

3

.2323232323232323333333232323

112112

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410注

1：矩阵乘法不满足交换律！

1

1

1

0

3

例：

A

，

B

，

1

1

0

2

4

1

2

7

A

B

，

但

B

A无意义！

1

2

7

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

1

0

例

A

，B

，

1

0

3

1

3

3

0

2

1

0

虽然

A

B

和B

A

都有意义，

但

A

B

B

A

.1001

A

B

，

B

A

1

0

3

.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410 1

1

1

1

例

A

1

1

，B

1

1

， 0

0

2

2A

B

，

B

A

，A

B

B

A

.

0

0

2

2

注

2

：两个非零矩阵的乘积可能是零矩阵！

由

AB

O推不出A

O或B

O

.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410注

3

：矩阵乘法不满足消去律！

1

1

1

2

3

5例：

A

、B

、

C

1

1

1

2

3

5

易知AB

AC

O，且

A

O，但

B

C

.所以

AB

AC

B

C

.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B4102.

矩阵乘法的性质⑴分配律：

左分配律：

A

B

C

AB

AC，右分配律：

B

C

A

BA

CA；⑵结合律：

A

B

C

A

B

C；Em

m

Am

n

A，A

E

A.m

n

n

n⑶对任意矩阵Am

n

：E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410设

A

aij

n

An

An

1

A

n

0

,

A0

E

,

A

0

.满足：

Al

Ak

Al

k

，(Ak

)l

Akl

k

0，l

0

.注意：(A

B)2

(A

B)(A

B)

A2

AB

BA

B2

.一般来说，(A

B)2

A2

2AB

B2

.n为方阵，

根据结合律，

我们定义：E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410设

f

x

an

xn

an

1xn

1

a1x

a0，

A

aij

n

n

则定义矩阵多项式：

为方阵,f

A

an

An

an

1An

1

a1A

a0E

.

1

1

例

f

x

2x

x

3，A

1

3

,

22

01x

3

2

2

1

3

6

则

f

A

2A2

A

3E

2

-

6

3

2

2

1=

.1

1

+3

1

00

1

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410例：

已知方阵A的各行元素之和都为常数a

，求证：方阵Am

的各行元素之和也为一个常数，证明：令

a11

A

a21

an1

1

1

，

1

a1n

，

a2n

ann

并求此常数aaa1222n2E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

a

1

a

a

1

a

.

a

1

1

a11

a12

1

=

a21

a22

a11

A

a21

a1n

a2n

ann

an1

an2

an1

1

aaaaa1n2n1222an2nnE6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410由矩阵乘法结合律，Am

Am

1

(A

)

aAm

1

am

.所以方阵Am

的各行元素之和都是常数

am

.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

a1nxn

b1

,

a11

a2nxn

b2

,

A

a21

amn

xn

bm

an1

a11x1

a12

x2

a21x1

a22

x2

x1

x

x2

xn

b1

b

b2

bm

aaan23.

方程组的矩阵形式a1n

a2n

amn

am1x1

am2x2

未知量

列向量常数项

列向量系数矩阵Ax

b1222E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵的运算E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410数与矩阵相乘矩阵与矩阵相乘

矩阵的转置方阵的行列式矩阵的加法

矩阵的线性运算E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵的转置E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B4101.矩阵的转置及性质定义

把矩阵A

的行换成同序数的列得到一个新矩阵，

叫做矩阵

A

的转置矩阵，

记为AT.

1

4

例：

设

A

2

3

，

则

A

.TTTTT3

23

23

23

2T6532

635241

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410矩阵转置的性质：(1)

AT

A;

由矩阵转置的定义即得.(2)

A

B

T

AT

BT

;

①等号两端的矩阵是同型矩阵.对应位置上的元素相等.TT(3)

A

B

B

A

.

②等号两端的矩阵TTTTTTTTTTTTT(4)

kA

T

kAT

.E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

2

0

例

:

已知

A

1

3解法1：

1

3

0

AB

14

3

1

7

1

1

02

1713.

1

7

1

2，B

14

3

，13

10

，求

AB

T.

1

3

1

17

10

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

1

7

1

例

:

已知

A

，B

，求

AB

.解法2：

1

4

2

210

17

AB

B

A

.TTTTTTTTTTTTTTTTTTTT

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410

1

1

3行3列矩阵

例

:

已知

2

，

1

，

A

，

求

An

.

1

1

1

1行1列矩阵

An

=(

T

)(

T

)

(

T

)

(

T

)(

T

)

(

T

)

T

1

1

1

TT

TT

解：

注意到

1

1

1

2

2，

1

=

2n

1

T

2n

1

T

2n

1

2

2

1

1

2.

1

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B4102

、对称阵与反对称阵

设方阵A

aij

n

n

.若矩阵A

满足AT

A

，则称A

为对称矩阵.直观来看，对称阵的元素关于对角线对称相等.

1

3

5

例

A

3

2

6

为对称阵；

5

6

8

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410若矩阵A

满足

AT

A

，则称A

为反对称矩阵.反对称阵的元素关于对角线异号;并且对角线上元素全为零。

0

2

4

例

B

2

0

6

为反对称阵.

4

6

0

E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410例

设列矩阵X

x1

,

x2

,

,

xn

T满足XT

X

1，E为n阶单位矩阵，H

E2XXT，证明H是对称阵，

且HHT

E.证明

因为HT

E

2XXT

ET

2XXT

E2XXT

H,

所以H是对称阵.HHT

H2

E

2XXT

E

4XXT

4

XT

XXXT

E

4XXT

4XXT

E.22TTTTTTTTTTTTTTTTTTTTTE6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B410方阵的行列式E6636B02012BD195C019CE06C16E30A4C03D0BEC2711A616D0819A1F3DE778FB8F41E803BC41BA963B46DCF0472B73A72FC0F2BE2AA532FE6901217257F3E41DBF2DB6715F2B3FBA75C24D6B4102

101

1

2

10位置不变），

1例

设

A

1

3

定义

由n阶方阵A的元素所构成的行列式（各元素