矩阵基础1-矩阵的基本知识
作者:
只是甲 | 来源:发表于
2022-05-20 11:12 被阅读0次
一. 矩阵的概念
1.1 运动会成绩记录
![](https://img.haomeiwen.com/i2638478/cfc54ca6caa13f6a.png)
image.png
我们通过对上图中的表格分析,可以知道每个系获得第一名、第二名、第三名、第四名以及第五名的个数。
然后每个名词对应的积分以及奖金都已经规定。
![](https://img.haomeiwen.com/i2638478/2dd49a3fdaeecf47.png)
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1.2 什么是矩阵
![](https://img.haomeiwen.com/i2638478/ccf481636ff31285.png)
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上一节中运动会案例刚好就是一个矩阵:
![](https://img.haomeiwen.com/i2638478/e4ef52ccc8b1f7c5.png)
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1.3 矩阵与向量
![](https://img.haomeiwen.com/i2638478/b6bb6f85caa2051d.png)
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复数可以分为两类数:实数、虚数。
实数包括有理数和无理数。
形如“a+bi”、“bi”(a、b∈R,并且b≠0)的复数都是虚数。
1.4 矩阵相等
![](https://img.haomeiwen.com/i2638478/c440b6460f1229e5.png)
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1.5 方阵
![](https://img.haomeiwen.com/i2638478/ef6676258eb781fc.png)
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1.6 单位矩阵
![](https://img.haomeiwen.com/i2638478/1c9dde7e8021c23e.png)
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1.7 负矩阵、上三角阵、下三角阵
![](https://img.haomeiwen.com/i2638478/7f1638c139b11728.png)
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1.8 对角方阵
![](https://img.haomeiwen.com/i2638478/a93dd502d14a44d8.png)
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1.9 零矩阵
![](https://img.haomeiwen.com/i2638478/6e0773de6b825b49.png)
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二. 矩阵的线性运算
矩阵 加法、减法和数乘:
减法其实就是加法的逆运算,此处省略
![](https://img.haomeiwen.com/i2638478/eafa0210ea0f237f.png)
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![](https://img.haomeiwen.com/i2638478/9d7b03444d1169b8.png)
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矩阵加法举例:
把100米与200米的名次汇总即可
![](https://img.haomeiwen.com/i2638478/fd985a6d7185d406.png)
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三. 矩阵的乘法
矩阵乘法公式:
![](https://img.haomeiwen.com/i2638478/d9a578fd5c66dfb9.png)
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![](https://img.haomeiwen.com/i2638478/6fbf8b4633ab8429.png)
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矩阵乘法举例:
把100米与200米的名次汇总后在 乘以 对应的积分与奖金
![](https://img.haomeiwen.com/i2638478/415c52d1f4c86bf3.png)
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例2
![](https://img.haomeiwen.com/i2638478/c420ad4a73a78fb5.png)
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例3
![](https://img.haomeiwen.com/i2638478/3143e0119d4addd7.png)
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例4
![](https://img.haomeiwen.com/i2638478/c65519d90f731717.png)
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矩阵乘法运算规则:
![](https://img.haomeiwen.com/i2638478/f52a73534b5d1a52.png)
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四. 矩阵的逆
4.1 矩阵转置
![](https://img.haomeiwen.com/i2638478/60b381a9d1531973.png)
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![](https://img.haomeiwen.com/i2638478/f9f4c0e45264db7b.png)
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4.2 对称矩阵
![](https://img.haomeiwen.com/i2638478/fd3c90229d71d901.png)
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![](https://img.haomeiwen.com/i2638478/49b7e68cde23c9b6.png)
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4.2 可逆矩阵
![](https://img.haomeiwen.com/i2638478/d71eec9458e67384.png)
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![](https://img.haomeiwen.com/i2638478/0edba6b87337f816.png)
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![](https://img.haomeiwen.com/i2638478/5cf00de6b901082d.png)
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五. 矩阵的行列式
行列式就是一组数排在一起。
5.1 分块矩阵
![](https://img.haomeiwen.com/i2638478/a34291b41dead930.png)
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分块矩阵的乘法:
![](https://img.haomeiwen.com/i2638478/c76b745609a1b097.png)
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分块方法:
![](https://img.haomeiwen.com/i2638478/2c7f39093d20301e.png)
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分块矩阵的转置:
![](https://img.haomeiwen.com/i2638478/c3bf105eb5aa81d4.png)
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5.2 二阶行列式
![](https://img.haomeiwen.com/i2638478/4791963e0b791bd5.png)
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5.3 三阶行列式
![](https://img.haomeiwen.com/i2638478/f772d46ea27fd5fa.png)
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5.4 n阶行列式
![](https://img.haomeiwen.com/i2638478/e027de04675d5e2b.png)
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5.5 特殊矩阵的行列式
![](https://img.haomeiwen.com/i2638478/0f7a8c2c67d74295.png)
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5.6 行列式的性质
![](https://img.haomeiwen.com/i2638478/ac46f7f4bd1bc025.png)
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![](https://img.haomeiwen.com/i2638478/beae2b2120d1b099.png)
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参考:
- http://www.dataguru.cn/article-4621-1.html
- https://www.zhihu.com/question/21082351/answer/1023650088
- https://baijiahao.baidu.com/s?id=1660241770276728313&wfr=spider&for=pc
本文标题:矩阵基础1-矩阵的基本知识
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