A= [1 0 0 0 1] [0 7 0 0 0] [0 0 3 0 0] [0 0 0 2 0] A^{T}= [1 0 0 0] [0 7 0 0] [0 0 3 0] [0 0 0 2] [1 0 0 0] A^{T}*A= [ 1 0 0 0 1] [ 0 49 0 0 0] [ 0 0 9 0 0] [ 0 0 0 4 0] [ 1 0 0 0 1] Eigenvalues & Eigenvectors of A^{T}*A [49, 9, 4, 2, 0] [(49, [ (0, 1, 0, 0, 0) ], 1), (9, [ (0, 0, 1, 0, 0) ], 1), (4, [ (0, 0, 0, 1, 0) ], 1), (2, [ (1, 0, 0, 0, 1) ], 1), (0, [ (1, 0, 0, 0, -1) ], 1)] V^T= [ 0 1 0 0 0] [ 0 0 1 0 0] [ 0 0 0 1 0] [ 1/2*sqrt(2) 0 0 0 1/2*sqrt(2)] [ 1/2*sqrt(2) 0 0 0 -1/2*sqrt(2)] A*A^{T}= [ 2 0 0 0] [ 0 49 0 0] [ 0 0 9 0] [ 0 0 0 4] Eigenvalues & Eigenvectors of A*A^{T} [49, 9, 4, 2] [(49, [ (0, 1, 0, 0) ], 1), (9, [ (0, 0, 1, 0) ], 1), (4, [ (0, 0, 0, 1) ], 1), (2, [ (1, 0, 0, 0) ], 1)] U= [ 0 0 0 1/2*sqrt(2)] [ 1 0 0 0] [ 0 1 0 0] [ 0 0 1 1/2*sqrt(2)] S= [ 7 0 0 0 0] [ 0 3 0 0 0] [ 0 0 2 0 0] [ 0 0 0 sqrt(2) 0] [ 0 0 0 0 0] sigma= [ 7 0 0 0 0] [ 0 3 0 0 0] [ 0 0 2 0 0] [ 0 0 0 sqrt(2) 0] U*(sigma)*Vt= [1/2*sqrt(2) 0 0 0 1/2*sqrt(2)] [ 0 7 0 0 0] [ 0 0 3 0 0] [1/2*sqrt(2) 0 0 2 1/2*sqrt(2)] A= [1 0 0 0 1] [0 7 0 0 0] [0 0 3 0 0] [0 0 0 2 0] A^{T}= [1 0 0 0] [0 7 0 0] [0 0 3 0] [0 0 0 2] [1 0 0 0] A^{T}*A= [ 1 0 0 0 1] [ 0 49 0 0 0] [ 0 0 9 0 0] [ 0 0 0 4 0] [ 1 0 0 0 1] Eigenvalues & Eigenvectors of A^{T}*A [49, 9, 4, 2, 0] [(49, [ (0, 1, 0, 0, 0) ], 1), (9, [ (0, 0, 1, 0, 0) ], 1), (4, [ (0, 0, 0, 1, 0) ], 1), (2, [ (1, 0, 0, 0, 1) ], 1), (0, [ (1, 0, 0, 0, -1) ], 1)] V^T= [ 0 1 0 0 0] [ 0 0 1 0 0] [ 0 0 0 1 0] [ 1/2*sqrt(2) 0 0 0 1/2*sqrt(2)] [ 1/2*sqrt(2) 0 0 0 -1/2*sqrt(2)] A*A^{T}= [ 2 0 0 0] [ 0 49 0 0] [ 0 0 9 0] [ 0 0 0 4] Eigenvalues & Eigenvectors of A*A^{T} [49, 9, 4, 2] [(49, [ (0, 1, 0, 0) ], 1), (9, [ (0, 0, 1, 0) ], 1), (4, [ (0, 0, 0, 1) ], 1), (2, [ (1, 0, 0, 0) ], 1)] U= [ 0 0 0 1/2*sqrt(2)] [ 1 0 0 0] [ 0 1 0 0] [ 0 0 1 1/2*sqrt(2)] S= [ 7 0 0 0 0] [ 0 3 0 0 0] [ 0 0 2 0 0] [ 0 0 0 sqrt(2) 0] [ 0 0 0 0 0] sigma= [ 7 0 0 0 0] [ 0 3 0 0 0] [ 0 0 2 0 0] [ 0 0 0 sqrt(2) 0] U*(sigma)*Vt= [1/2*sqrt(2) 0 0 0 1/2*sqrt(2)] [ 0 7 0 0 0] [ 0 0 3 0 0] [1/2*sqrt(2) 0 0 2 1/2*sqrt(2)] |
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