matlab lud矩阵分解,MIT线性代数总结笔记——LU分解

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MIT线性代数总结笔记——LU分解
矩阵分解
矩阵分解(Matrix Factorizations)就是将一个矩阵用两个以上的矩阵相乘的等式来表达。而矩阵乘法涉及到数据的合成(即将两个或多个线性变换的效果组合成一个矩阵)，因此可以说，矩阵分解是对数据的一种分析。
矩阵分解
LU分解
分解形式
<img alt="math?formula=A%3DLU" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-b6a2e62d9ecbb0579bb6eaf88c5898a4">
(
<img alt="math?formula=L" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-bfc6bff3b32596a9a7ce0299fe4f3a95">代表下三角矩阵，
<img alt="math?formula=U" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-be8039ad246c79c8bcb10284fa53de62">代表上三角矩阵)
目的
提高计算效率
前提
(1)矩阵是方阵(
<img alt="math?formula=LU" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-5aff5c8cc6c6a2742a4218c621e4b0ad">分解主要是针对方阵)；
(2)矩阵是可逆的，也就是该矩阵是满秩矩阵，每一行都是独立向量；
(3)消元过程中没有
<img alt="math?formula=0" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-236b3a5e048b42fd14f790093074602e">主元出现，也就是消元过程中不能出现行交换的初等变换。
LU分解
分解形式
<img alt="math?formula=A%20%3D%20LU%20%3D%20%5Cleft%5B%5Cbegin%7Bmatrix%7D%201%20%26%200%20%26%200%20%26%200%20%5C%5C%20l_%7B21%7D%20%26%201%20%26%200%20%26%200%20%5C%5C%20l_%7B31%7D%20%26%20l_%7B32%7D%20%26%201%20%26%200%20%5C%5C%20l_%7B41%7D%20%26%20l_%7B42%7D%20%26%20l_%7B43%7D%20%26%201%5Cend%7Bmatrix%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Bmatrix%7D%20u_%7B11%7D%20%26%20u_%7B12%7D%20%26%20u_%7B13%7D%20%26%20u_%7B14%7D%20%5C%5C%200%20%26%20u_%7B22%7D%20%26%20u_%7B23%7D%20%26%20u_%7B24%7D%20%5C%5C%200%20%26%200%20%26%20u_%7B33%7D%20%26%20u_%7B34%7D%20%5C%5C%200%20%26%200%20%26%200%20%26%20u_%7B44%7D%5Cend%7Bmatrix%7D%5Cright%5D" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-7842f255da0d82bc266e9b19b0b02f13">
简述
<img alt="math?formula=LU" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-5aff5c8cc6c6a2742a4218c621e4b0ad">分解常用于求解工业和商业问题中的序列方程。它是最常见的求解线性系统 
<img alt="math?formula=Ax%3Db" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-54747b8b9c6171979f77c731ac8e6b49"> 的方法，主要思路是：把
<img alt="math?formula=A" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-7802ad344253f09ef186a7bed8f35c5e"> 分解成一个下三角矩阵(Lower Triangular Matrix)和一个上三角矩阵(Upper Triangular Matrix)，简称
<img alt="math?formula=LU" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-5aff5c8cc6c6a2742a4218c621e4b0ad">。分解后，等式
<img alt="math?formula=Ax%3Db" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-54747b8b9c6171979f77c731ac8e6b49">可以写成
<img alt="math?formula=L(Ux)%3Db" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-d7b9c2eee832271eb8c95f5760f4d4d1">，这样我们令
<img alt="math?formula=y%3DUx" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-a67a6e2e048c6fbd1288e30a776bfb16">，这样就可以通过分开求解两个等式来得到
<img alt="math?formula=x" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-89630d1bb0c18f78d0a1401760492dfa">：
<img alt="math?formula=Ly%3Db%20%5C%5C%20Ux%3Dy" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-927edf1470086b5c0612ac04d117c1fd">
即可以先通过
<img alt="math?formula=Ly%3Db" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-ce21b6d934af6a93ce92738efd3ce51b">求出
<img alt="math?formula=y" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-1f97f2a5da40c6624c56618ceb797248">，最后再用
<img alt="math?formula=Ux%3Dy" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-a8b89c6785849a5df9dfa2e5ec7ec6bd">，求出
<img alt="math?formula=x" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-89630d1bb0c18f78d0a1401760492dfa">。
本质上，
<img alt="math?formula=LU" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-5aff5c8cc6c6a2742a4218c621e4b0ad">分解是高斯消元法的一种表达方式，为了更好地理解
<img alt="math?formula=LU" src="https://beijingoptbbs.oss-cn-beijing.aliyuncs.com/cs/5606289-5aff5c8cc6c6a2742a4218c621e4b0ad">分解，我们需要解释一下高斯消元法。

matlab lud矩阵分解,MIT线性代数总结笔记——LU分解

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