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\begin_body
\begin_layout Section
矩阵的概念
\end_layout
\begin_layout Subsection
矩阵的概念
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
引言
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
线性方程组求解的一般性理论;
\end_layout
\begin_layout Itemize
关系型数据库的表格形数据;
\end_layout
\begin_layout Itemize
图论中顶点间连接情况的邻接矩阵;
\end_layout
\begin_layout Itemize
通信纠错问题与纠错码.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 3
status open
\begin_layout Plain Layout
allowframebreaks
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
矩阵的概念
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Definition
由
\begin_inset Formula $m\times n$
\end_inset
个数
\begin_inset Formula $a_{ij}$
\end_inset
(
\begin_inset Formula $i=1,2,\cdots,m$
\end_inset
;
\begin_inset Formula $j=1,2,\cdots,n$
\end_inset
) 排成的
\begin_inset Formula $m$
\end_inset
行
\begin_inset Formula $n$
\end_inset
列的数表
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{-2mm}
\end_layout
\end_inset
\begin_inset Formula
\[
\begin{array}{llll}
a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{array}
\]
\end_inset
称为
\begin_inset Formula $m$
\end_inset
行
\begin_inset Formula $n$
\end_inset
列
\series bold
矩阵
\series default
, 简称
\series bold
\begin_inset Formula $m\times n$
\end_inset
\series default
矩阵.
为表示它是一个整体, 总是加一个括弧 (或方括号), 并用大写字母表示它, 记为
\begin_inset Formula
\[
A=\begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix}
\]
\end_inset
这
\begin_inset Formula $m\times n$
\end_inset
个数称为矩阵
\begin_inset Formula $A$
\end_inset
的元素,
\begin_inset Formula $a_{ij}$
\end_inset
称为矩阵
\begin_inset Formula $A$
\end_inset
的第
\begin_inset Formula $i$
\end_inset
行第
\begin_inset Formula $j$
\end_inset
列元素.
一个
\begin_inset Formula $m\times n$
\end_inset
矩阵
\begin_inset Formula $A$
\end_inset
也可简记为
\begin_inset Formula
\[
A=A_{m\times n}=(a_{ij})_{m\times n}\text{ 或 }A=(a_{ij}).
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Definition
元素是实数的矩阵称为
\series bold
实矩阵
\series default
, 元素是复数的矩阵称为
\series bold
复矩阵
\series default
, 本课程中的矩阵都指实矩阵 (除非有特殊说明).
\end_layout
\begin_layout Definition
所有元素均为零的矩阵称为
\series bold
零矩阵
\series default
, 记为
\begin_inset Formula $O$
\end_inset
.
\end_layout
\begin_layout Definition
所有元素均为非负数的矩阵称为
\series bold
非负矩阵
\series default
.
\end_layout
\begin_layout Definition
若矩阵
\begin_inset Formula $A=(a_{ij})$
\end_inset
的行数与列数都等于
\begin_inset Formula $n$
\end_inset
, 则称
\begin_inset Formula $A$
\end_inset
为
\series bold
\begin_inset Formula $n$
\end_inset
阶方阵
\series default
, 记为
\begin_inset Formula $A_{n}$
\end_inset
.
\end_layout
\begin_layout Definition
如果两个矩阵具有相同的行数与相同的列数, 则称这两个矩阵为
\series bold
同型矩阵
\series default
.
\end_layout
\begin_layout Definition
如果矩阵
\begin_inset Formula $A,B$
\end_inset
为同型矩阵, 且对应元素均相等, 则称矩阵
\begin_inset Formula $A$
\end_inset
与矩阵
\begin_inset Formula $B$
\end_inset
相等, 记为
\begin_inset Formula $A=B$
\end_inset
.
\end_layout
\begin_layout Example
设
\begin_inset Formula $A=\begin{bmatrix}1 & 2-x & 3\\
2 & 6 & 5z
\end{bmatrix}$
\end_inset
,
\begin_inset Formula $B=\begin{bmatrix}1 & x & 3\\
y & 6 & z-8
\end{bmatrix}$
\end_inset
, 已知
\begin_inset Formula $A=B$
\end_inset
, 求
\begin_inset Formula $x,y,z$
\end_inset
.
\end_layout
\begin_layout Solution*
因为
\begin_inset Formula $2-x=x$
\end_inset
,
\begin_inset Formula $2=y$
\end_inset
,
\begin_inset Formula $5z=z-8$
\end_inset
, 所以
\begin_inset Formula
\[
x=1,\ y=2,\ z=-2.
\]
\end_inset
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 4
status open
\begin_layout Plain Layout
矩阵概念的应用
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
广义逆矩阵求解超定方程;
\end_layout
\begin_layout Itemize
人工智能领域中的神经网络, 卷积神经网络 (CNN), 循环神经网络 (RNN), 深度神经网络 (DNN), 长短期记忆 (LSTM), 强化学习等模型的简化
书写;
\end_layout
\begin_layout Itemize
图论领域中的应用, 如最短路径问题;
\end_layout
\begin_layout Itemize
运筹学中的应用, 如线性规划, 整数规划;
\end_layout
\begin_layout Itemize
数论加密算法领域;
\end_layout
\begin_layout Itemize
时间序列分析, 如应用于投资领域.
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Frame
\begin_inset Argument 3
status open
\begin_layout Plain Layout
allowframebreaks
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
几种特殊矩阵
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Itemize
只有一行的矩阵
\begin_inset Formula
\[
A=\begin{pmatrix}a_{1} & a_{2} & \cdots & a_{n}\end{pmatrix}
\]
\end_inset
称为
\series bold
行矩阵
\series default
或
\series bold
行向量
\series default
.
为避免元素间的混淆, 行矩阵也记作
\begin_inset Formula
\[
A=\left(a_{1},a_{2},\cdots,a_{n}\right).
\]
\end_inset
\end_layout
\begin_layout Itemize
只有一列的矩阵
\begin_inset Formula
\[
B=\begin{bmatrix}b_{1}\\
b_{2}\\
\vdots\\
b_{m}
\end{bmatrix}
\]
\end_inset
称为
\series bold
列矩阵
\series default
或
\series bold
列向量
\series default
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $n$
\end_inset
阶方阵
\begin_inset Formula
\[
\begin{bmatrix}\lambda_{1} & 0 & \cdots & 0\\
0 & \lambda_{2} & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & \lambda_{n}
\end{bmatrix}
\]
\end_inset
称为
\series bold
\begin_inset Formula $n$
\end_inset
阶对角矩阵
\series default
, 对角矩阵也记为
\begin_inset Formula
\[
A=\mathrm{diag}\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{n}\right).
\]
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $n$
\end_inset
阶方阵
\begin_inset Formula
\[
\begin{bmatrix}1 & 0 & \cdots & 0\\
0 & 1 & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & 1
\end{bmatrix}
\]
\end_inset
称为
\series bold
\begin_inset Formula $n$
\end_inset
阶单位矩阵
\series default
,
\begin_inset Formula $n$
\end_inset
阶单位矩阵也记为
\begin_inset Formula
\[
E=E_{n},\quad(\text{或 }I=I_{n}).
\]
\end_inset
\end_layout
\begin_layout Itemize
当一个
\begin_inset Formula $n$
\end_inset
阶对角矩阵
\begin_inset Formula $A$
\end_inset
的对角元素全部相等且等于某一数
\begin_inset Formula $a$
\end_inset
时, 称
\begin_inset Formula $A$
\end_inset
为
\series bold
\begin_inset Formula $n$
\end_inset
阶数量矩阵
\series default
, 即
\begin_inset Formula
\[
A=\begin{bmatrix}a & 0 & \cdots & 0\\
0 & a & \cdots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \cdots & a
\end{bmatrix}
\]
\end_inset
\end_layout
\end_deeper
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Subsection
例题
\end_layout
\begin_layout Frame
\begin_inset Argument 3
status open
\begin_layout Plain Layout
allowframebreaks
\end_layout
\end_inset
\begin_inset Argument 4
status open
\begin_layout Plain Layout
例题
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Example
甲、乙、丙、丁、戊五人各从图书馆借来一本小说, 他们约定读完后互相交换, 这五本书的厚度以及他们五人的阅读速度差不多, 因此, 五人总是同时交换书,
经四次交换后, 他们五人读完了这四本书, 现已知:
\end_layout
\begin_deeper
\begin_layout Enumerate
甲最后读的书是乙读的第二本书;
\end_layout
\begin_layout Enumerate
丙最后读的书是乙读的第四本书;
\end_layout
\begin_layout Enumerate
丙读的第二本书甲在一开始就读了;
\end_layout
\begin_layout Enumerate
丁最后读的书是丙读的第三本;
\end_layout
\begin_layout Enumerate
乙读的第四本书是戊读的第三本书;
\end_layout
\begin_layout Enumerate
丁第三次读的书是丙一开始读的那本书.
\end_layout
\begin_layout Standard
试根据以上情况说出丁第二次读的书是谁最先读的书?
\end_layout
\end_deeper
\begin_layout Solution*
设甲、乙、丙、丁、戊最后读的书的代号依次为
\begin_inset Formula $A$
\end_inset
,
\begin_inset Formula $B$
\end_inset
,
\begin_inset Formula $C$
\end_inset
,
\begin_inset Formula $D$
\end_inset
,
\begin_inset Formula $C$
\end_inset
,
\begin_inset Formula $E$
\end_inset
, 则根据题设条件可以列出下列初始矩阵为
\end_layout
\begin_layout Solution*
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\backslash
begin{bNiceArray}{ccccc}[first-row,first-col]
\end_layout
\begin_layout Plain Layout
&
\backslash
text{甲} &
\backslash
text{乙} &
\backslash
text{丙} &
\backslash
text{丁} &
\backslash
text{戊}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
1 & x & & y & &
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
2 & & A & x & &
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
3 & & & D & y & C
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
4 & & C & & &
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
5 & A & B & C & D & E
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceArray}
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
\end_layout
\begin_layout Solution*
上述矩阵中的
\begin_inset Formula $x,y$
\end_inset
表示尚未确定的书名代号.
两个
\begin_inset Formula $x$
\end_inset
代表同一本书, 两个
\begin_inset Formula $y$
\end_inset
代表另外的同一本书.
由题意知, 经
\begin_inset Formula $5$
\end_inset
次阅读后乙将五本书全都阅读了, 则从上述矩阵可以看出, 乙第 3 次读的书不可能是
\begin_inset Formula $A,B$
\end_inset
或
\begin_inset Formula $C$
\end_inset
.
另外由于丙在第
\begin_inset Formula $3$
\end_inset
次阅读的是
\begin_inset Formula $D$
\end_inset
, 所以乙第
\begin_inset Formula $3$
\end_inset
次读的书也不可能是
\begin_inset Formula $D$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Solution*
因此, 乙第
\begin_inset Formula $3$
\end_inset
次读的书是
\begin_inset Formula $E$
\end_inset
, 从而乙第
\begin_inset Formula $1$
\end_inset
次读的书是
\begin_inset Formula $D$
\end_inset
.
同理可推出甲第
\begin_inset Formula $3$
\end_inset
次读的书是
\begin_inset Formula $B$
\end_inset
.
因此上述矩阵中的
\begin_inset Formula $y$
\end_inset
为
\begin_inset Formula $A$
\end_inset
,
\begin_inset Formula $x$
\end_inset
为
\begin_inset Formula $E$
\end_inset
.
由此可得到各个人的阅读顺序, 如下述矩阵所示:
\end_layout
\begin_layout Solution*
\begin_inset ERT
status open
\begin_layout Plain Layout
$$
\backslash
begin{bNiceArray}{ccccc}[first-row,first-col]
\end_layout
\begin_layout Plain Layout
&
\backslash
text{甲} &
\backslash
text{乙} &
\backslash
text{丙} &
\backslash
text{丁} &
\backslash
text{戊}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
1 & E & D & A & C & B
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
2 & C & A & E & B & D
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
3 & B & E & D & A & C
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
4 & D & C & B & E & A
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
5 & A & B & C & D & E
\end_layout
\begin_layout Plain Layout
\backslash
end{bNiceArray}
\end_layout
\begin_layout Plain Layout
$$
\end_layout
\end_inset
\end_layout
\begin_layout Solution*
由此矩阵知, 丁第 2 次读的书是戊一开始读的那一本书.
\end_layout
\end_deeper
\begin_layout Frame
\end_layout
\end_body
\end_document
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