2L

Data compression using reduced rank approximations
- the rank of a matix specifies the number of linearly independent columns (or rows)
- a measure of redundancy, matrix of low rank has a large amount of redundancy
* a matrix B of rank one, columns are all multiples of one another
- u as a basis in R^m, then B = [v_1u, ..., v_nu] = uv^T
- m+n elements vs. mxn

- Matrix B is the best rank one approximation to A if the error matrix B - A has the minimal Frobenius norm.
- Frobenius norm of matrix X: |X| = \sqrt{\sigma_{ij} x_{ij}^2} (note: square root)
- This norm is just the Eculidean norm \|x\|_2 of the matrix considered as a vector in R^{mn}
~~~~~~~~~ inner product related ~~~~~~~~~~~~~~~~
* Inner product of two matrices X\cdot Y = \sigma_{ij}x_{ij}y_{ij} (pointwise multiplication)
- |X|^2 = X\cdot X (squared Frobenius norm = self inner product)
- inner product of matrices can be viewed in three ways
=1= sum of the inner product of corresponding rows
=2= sum of the inner product of corresponding columns
=3= sum of corresponding entries
* For rank one matrices xy^T, uv^T
xy^T \cdot uv^T
= [xy_1, ..., xy_n] \cdot [uv_1, ..., uv_n] ~~~~~ write in columns form
= \sigma_i xy_i \cdot uv_i ~~~~~~ inner product of corresponding cols
= \sigma_i (x \cdot u)(y_iv_i) ~~~~~~ move scalar out
= (x \cdot u)(y \cdot v) ~~~~~~ write in vector inner product form

- Matrix A in SVD form

- Frobenius norm of matrix A ( the 4th equality => 4L )

matlab mathworks moler eigs.pdf
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定义：eigenvalue a, eigenvector x
Ax = ax
定义：singular value o, 2 sigular vector u,v
Av = ou (1)
AHu = ov (2)
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H:hermitian
由(2)得AAHu = oAv = o2u，也就是AAHu = o2u，即
o2是AAH的特征值，u是对应的的特征向量
同理
由(1)得AHAv = oAHu = o2v，也就是AHAv = o2v，即
o2是AHA的特征值，v是对应的特征向量
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