Single Value Decomposition (SVD) in action:
Definition: SVD divides a matrix A into three matrices: UΣv^t.
- U: M x M unitary matrix (left singular vectors)
- Σ: m x n diagonal matrix with singular values on the diagonal
- V: N x N unitary matrix (right singular vectors)
Calculation of SVD (example):
Matrix A:
A = [4 7]
[11 2]
Calculate A^T and AA^T:
A^t A = [65 58]
[58 125]
AA^t = [137 78]
[78 53]
Find the eigenvalues and eigenvectors of A^T and AA^T:
For A^T A:
Eigenvalues: λ1 = 178, λ2 = 12
Eigenvectors: v1 = [0.71 0.71], v2 = [-0.71 0.71]
For AA^T:
Eigenvalues: λ1 = 190, λ2 = 0
Eigenvectors: u1 = [0.85 0.53], u2 = [-0.53 0.85]
Calculate the singular values σi:
σ1 = √λ1 = √190
σ2 = √λ2 = 0
Construct U, Σ, V:
U = [0.85 -0.53]
[0.53 0.85]
Σ = [190 0]
[ 0 0 ]
V = [0.71 -0.71]
[0.71 0.71]
SVD of matrix A:
A = uΣv^t = [0.85 -0.53] [190 0] [0.71 -0.71]
[0.53 0.85] [ 0 0] [0.71 0.71]
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