9주차 강의

# 예제 1 A = matrix([[sqrt(3), 2], [0, sqrt(3)]]) B = A.transpose()*A eig = B.eigenvalues() sv = [sqrt(i) for i in eig] # 특이값 구하기 print(B.eigenvectors_right()) # A^T * A 의 고유벡터 구하기, A의 right singular vector들 

[(9, [(1, sqrt(3))], 1), (1, [(1, -1/3*sqrt(3))], 1)]

[(9, [(1, sqrt(3))], 1), (1, [(1, -1/3*sqrt(3))], 1)]

G = matrix([[1, sqrt(3)], [1, -1/3*sqrt(3)]]) Vh = matrix([1/G.row(i).norm()*G.row(j) for j in range(0,2)]) # V의 전치행렬 Vh = Vh.simplify() # V의 전치행렬 표현 print Vh print U = matrix([A*Vh.row(j)/sv[j] for j in range(0, 2)]).transpose() # U print U # U의 열들을 A의 left singular vector들 print S = diagonal_matrix(sv) print S print print U*S*Vh 

[1/2*sqrt(3)         3/2]
[1/2*sqrt(3)        -1/2]

[         3/2          1/2]
[ 1/2*sqrt(3) -1/2*sqrt(3)]

[3 0]
[0 1]

[5/2*sqrt(3)        13/2]
[        3/2 5/2*sqrt(3)]

[1/2*sqrt(3)         3/2]
[1/2*sqrt(3)        -1/2]

[         3/2          1/2]
[ 1/2*sqrt(3) -1/2*sqrt(3)]

[3 0]
[0 1]

[5/2*sqrt(3)        13/2]
[        3/2 5/2*sqrt(3)]

# 예제 2 - full column rank인 A의 pseudo-inverse 구하기 A = matrix([[1,1],[0,1],[1,0]]) print 'A=' print A print print 'pseudo-inverse of A=' print (A.transpose()*A).inverse()*A.transpose() # pseudo-inverse 구하기 

A=
[1 1]
[0 1]
[1 0]

pseudo-inverse of A=
[ 1/3 -1/3  2/3]
[ 1/3  2/3 -1/3]

A=
[1 1]
[0 1]
[1 0]

pseudo-inverse of A=
[ 1/3 -1/3  2/3]
[ 1/3  2/3 -1/3]

# 예제 4 - 최소제곱직선 구하기 A = matrix([[1,0],[1,1],[1,2],[1,3]]) b = vector([1,3,4,4]) print (A.transpose()*A).inverse()*A.transpose()*b 

(3/2, 1)

(3/2, 1)