LA Ch8 Exs Prob 14 by 김지윤

383 days ago by jykim

#1 A=matrix(4, 3, [1,0,0,0,0,4,0,3,0,0,0,3]) B=(A.transpose())*A print "A'A=" print B print eig=B.eigenvalues() print "eigenvalues of A'A=" print eig print print "eigenvectors of A'A=" print B.right_eigenvectors() v1=B.right_eigenvectors()[0][1][0]/B.right_eigenvectors()[0][1][0].norm() v2=B.right_eigenvectors()[1][1][0]/B.right_eigenvectors()[1][1][0].norm() v3=B.right_eigenvectors()[2][1][0]/B.right_eigenvectors()[2][1][0].norm() print V=column_matrix([v1,v2,v3]) print "V=" print V 
       
A'A=
[ 1  0  0]
[ 0  9  0]
[ 0  0 25]

eigenvalues of A'A=
[25, 9, 1]

eigenvectors of A'A=
[(25, [
(0, 0, 1)
], 1), (9, [
(0, 1, 0)
], 1), (1, [
(1, 0, 0)
], 1)]

V=
[0 0 1]
[0 1 0]
[1 0 0]
A'A=
[ 1  0  0]
[ 0  9  0]
[ 0  0 25]

eigenvalues of A'A=
[25, 9, 1]

eigenvectors of A'A=
[(25, [
(0, 0, 1)
], 1), (9, [
(0, 1, 0)
], 1), (1, [
(1, 0, 0)
], 1)]

V=
[0 0 1]
[0 1 0]
[1 0 0]
#2 B1=A*(A.transpose()) print "AA'=" print B1 print eig1=B1.eigenvalues() print "eigenvalues of AA'=" print eig1 print print "eigenvectors of AA'=" print B1.right_eigenvectors() u1=B1.right_eigenvectors()[0][1][0]/B1.right_eigenvectors()[0][1][0].norm() u2=B1.right_eigenvectors()[1][1][0]/B1.right_eigenvectors()[1][1][0].norm() u3=B1.right_eigenvectors()[2][1][0]/B1.right_eigenvectors()[2][1][0].norm() u4=B1.right_eigenvectors()[3][1][0]/B1.right_eigenvectors()[3][1][0].norm() print U=column_matrix([u1,u2,u3,u4]) print "U=" print U 
       
AA'=
[ 1  0  0  0]
[ 0 16  0 12]
[ 0  0  9  0]
[ 0 12  0  9]

eigenvalues of AA'=
[25, 9, 1, 0]

eigenvectors of AA'=
[(25, [
(0, 1, 0, 3/4)
], 1), (9, [
(0, 0, 1, 0)
], 1), (1, [
(1, 0, 0, 0)
], 1), (0, [
(0, 1, 0, -4/3)
], 1)]

U=
[   0    0    1    0]
[ 4/5    0    0  3/5]
[   0    1    0    0]
[ 3/5    0    0 -4/5]
AA'=
[ 1  0  0  0]
[ 0 16  0 12]
[ 0  0  9  0]
[ 0 12  0  9]

eigenvalues of AA'=
[25, 9, 1, 0]

eigenvectors of AA'=
[(25, [
(0, 1, 0, 3/4)
], 1), (9, [
(0, 0, 1, 0)
], 1), (1, [
(1, 0, 0, 0)
], 1), (0, [
(0, 1, 0, -4/3)
], 1)]

U=
[   0    0    1    0]
[ 4/5    0    0  3/5]
[   0    1    0    0]
[ 3/5    0    0 -4/5]
#3 sv=[sqrt(i) for i in eig] d1=sv[0] d2=sv[1] d3=sv[2] print "singular values =", (d1,d2,d3) print S=matrix(4,3,[d1,0,0,0,d2,0,0,0,d3,0,0,0]) print "S=" print S 
       
singular values = (5, 3, 1)

S=
[5 0 0]
[0 3 0]
[0 0 1]
[0 0 0]
singular values = (5, 3, 1)

S=
[5 0 0]
[0 3 0]
[0 0 1]
[0 0 0]
#4 C=U*S*V1 print "A=UST'=" print C 
       
A=UST'=
[1 0 0]
[0 0 4]
[0 3 0]
[0 0 3]
A=UST'=
[1 0 0]
[0 0 4]
[0 3 0]
[0 0 3]