# LA.Ch8.Ex14.NEW.SC.Kim

## 2024 days ago by seongchan

A = matrix(4,5, [1,0,0,0,1,0,7,0,0,0,0,0,3,0,0,0,0,0,2,0]) B = A.transpose() C= B*A D= A*B eig_C = C.eigenvalues() eig_D = D.eigenvalues() F=matrix([[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,1],[1,0,0,0,-1]]) G=matrix([1/F.row(j).norm()*F.row(j) for j in range(0,5)]) Vt=G.simplify() H=matrix([[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,1]]) J=matrix([1/H.row(j).norm()*H.row(j) for j in range(0,4)]) U=J.simplify().transpose() sv=[sqrt(i) for i in eig_C] S=diagonal_matrix(sv) K=matrix([[7,0,0,0,0],[0,3,0,0,0],[0,0,2,0,0],[0,0,0,sqrt(2),0]]) print "A=" print A print print "A^{T}=" print B print print "A^{T}*A=" print C print print "Eigenvalues & Eigenvectors of A^{T}*A" print eig_C print C.eigenvectors_right() print print "V^T=" print Vt print print "A*A^{T}=" print D print print "Eigenvalues & Eigenvectors of A*A^{T}" print eig_D print D.eigenvectors_right() print print "U=" print U print print "S=" print S print "sigma=" print K print print "U*(sigma)*Vt=" print U*K*Vt
 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)]