Today we have learned the Matrix Factorization, and I want to record my study notes. Some kownledge which I have learned before is forgot...（呜呜）
1.Terminology
单位矩阵：identity matrix
特征值：eigenvalues
特征向量：eigenvectors
矩阵的秩：rank
对角矩阵：diagonal matrix
对角化矩阵：Diagonalizing a Matrix
矩阵分解：matrix factorization
奇异值分解：SVD(singular value decomposition)
2.Basic kowledge
<1>Eigenvalues and Eigenvectors
What: The basic equation is 
, the x is the eigenvector of A and the lamda is the eigenvalue of A.
How: We can transpose this equation as 
(I is identity matrix), so we will kown the 
. We can calculate the eigenvalues and then get the eigenvectors.
<2>Diagonalizing a Matrix
What: (大家都知道，但是特别注意下形如下面的两种矩阵也是对角矩阵)
How: 
,lamada is the eigenvalue of A, and the column of S is the eigenvector of A. Like follow:
<3>rank
- The Rank and the Row Reduced Form （注：我们知道矩阵秩的定义或者求法有很多种，这里说的是行/列最简形矩阵的行/列数即为矩阵的秩，或者就是矩阵的最大非零r阶子式，则r称为矩阵的秩，即R(A)=r ）
- If the rank of matrix A(n*n) is r, what it's mean. 矩阵A的列向量或者行向量只有r个是非线性相关的，其他的n-r个向量是无价值的。（这个很重要，下面矩阵分解将会用到，自己的感悟不会用英文表达了，用中文。。。）
<4>singular value
3. Matrix Factorization
What:
（注：这里注意，当k>=m或k>=n且矩阵U或者V是满秩的，矩阵无法分解）
Why: We can use this to
- Image Recovery
(recovery this image)
-
Recommendation
(evaluate the ?)
-
and so on
How:
<1>Matrix completation
（注：在这里我们需要做出假设，即矩阵是低秩。为什么呢？在上面基础概念中我们说到了矩阵秩所代表的意义，那么这里我们假设矩阵是低秩的，则说明矩阵的列向量只有少数几列是真正重要的，其他的都是和这几列线性相关的，那么我们就可以通过这种线性相关来补全残缺的矩阵）
<2>SVD is one of the methods of matrix factorization, we will introduce this method below.
We have discussed the Diagonalizing a Matrix,but when A is any m by n matrix, square or rectangular. Its rank is r. We will diagonalize this A, but not by 
. The eigenvectors in S have three big problems: They are usually not orthogonal, there are not always enough eigenvectors, and 
requires A to be square. The singular vectors of A solve all those problems in a perfect way.
singular value:（这里有一篇博文说的很好，推荐下）
通过上面的方法，能够对矩阵X做出一定的降秩，但是如果这里d与m或者n较为接近，那么降秩的效果就不明显，所以我们使用一种近似的策略，如下：
这样我们就是实现了将一个矩阵A，经过SVD近似，转换成一个同维度的矩阵A*，但是它的秩远远低于A
to be continued...