1-1 Cauchy–Green deformation tensor
9
More details about the polar decomposition related to the singular value decomposition can be found in
the classical text by N. J. Highham (1986), C. Kenney and A. J. Laub (1991), and T. C. T. Ting (1985).
Example 1.4 is a numerical example for singular value decomposition and polar decomposition.
Example 1.4 (Singular value decomposition, polar decomposition).
Let there be given the Jacobi matrix J and the product matrices JJ
∗ and J
∗ J such that the left and
right characteristic equations of eigenvalues read
J =
5 2
−1 7
, JJ
∗ =
29 9
9 50
, J
∗ J =
26 3
3 53
,
(1.33)
|JJ
∗
− λI 2 | =
|J
∗ J − λI 2 | =
=
29 − λ 9
9 50− λ
=
=
26 − λ 3
3 53− λ
=
= λ
2
− 79λ + 1369 =
= λ
2
− 79λ + 1369 =
= 0 ,
= 0 ,
(1.34)
I := tr [JJ
∗ ] = tr [J
∗ J] = 79 , II := det [JJ
∗ ] = det [J
∗ J] = 1369 ,
(1.35)
λ 1 = 53.329 317 , σ 1 =
√
λ 1 = 7.302 692 ,
λ 2 = 25.670 683 , σ 2 =
√
λ 2 = 5.066 624 .
(1.36)
The left eigenspace is spanned by the left eigencolumns (u 1 , u 2 ), the right eigenspace by the right
eigencolumns (v 1 , v 2 ), namely
(JJ
∗
− λ 1 I 2 )u 1 = 0 ,
(J
∗ J − λ 1 I 2 )v 1 = 0 ,
(JJ
∗
− λ 2 I 2 )u 2 = 0 ,
(J
∗ J − λ 2 I 2 )v 2 = 0 ,
(1.37)
or
−24.329 317
9
9
−3.329 317
u 11
u 21
= 0 ,
−27.329 317
3
3
−0.329 317
v 11
v 21
= 0 ,
3.329 317
9
9
24.329 317
u 12
u 22
= 0 ,
0.329 317
3
3
27.329 317
v 12
v 22
= 0 .
(1.38)
Note that the matrices JJ
∗
− λI 2 and J
∗ J − λI 2 have only rank one. Accordingly, in order to solve the
homogenous linear equations uniquely, we need an additional constraint. Conventionally, this problem
is solved by postulating normalized eigencolumns, namely
u
2
11 + u
2
21 = 1 , u
2
12 + u
2
22 = 1 , v
2
11 + v
2
21 = 1 , v
2
12 + v
2
22 = 1 ,
u 1 = u 2 = 1 ,
v 1 = v 2 = 1 .
(1.39)
The left eigencolumns, which are here denoted as (u 1 , u 2 ), are constructed from the following system
of equations:
−24.329 317u 11 + 9u 21 = 0 , +3.329 317u 12 + 9u 22 = 0 ,
u
2
11 + u
2
21 = 1 , u
2
12 + u
2
22 = 1 .
(1.40)
9
More details about the polar decomposition related to the singular value decomposition can be found in
the classical text by N. J. Highham (1986), C. Kenney and A. J. Laub (1991), and T. C. T. Ting (1985).
Example 1.4 is a numerical example for singular value decomposition and polar decomposition.
Example 1.4 (Singular value decomposition, polar decomposition).
Let there be given the Jacobi matrix J and the product matrices JJ
∗ and J
∗ J such that the left and
right characteristic equations of eigenvalues read
J =
5 2
−1 7
, JJ
∗ =
29 9
9 50
, J
∗ J =
26 3
3 53
,
(1.33)
|JJ
∗
− λI 2 | =
|J
∗ J − λI 2 | =
=
29 − λ 9
9 50− λ
=
=
26 − λ 3
3 53− λ
=
= λ
2
− 79λ + 1369 =
= λ
2
− 79λ + 1369 =
= 0 ,
= 0 ,
(1.34)
I := tr [JJ
∗ ] = tr [J
∗ J] = 79 , II := det [JJ
∗ ] = det [J
∗ J] = 1369 ,
(1.35)
λ 1 = 53.329 317 , σ 1 =
√
λ 1 = 7.302 692 ,
λ 2 = 25.670 683 , σ 2 =
√
λ 2 = 5.066 624 .
(1.36)
The left eigenspace is spanned by the left eigencolumns (u 1 , u 2 ), the right eigenspace by the right
eigencolumns (v 1 , v 2 ), namely
(JJ
∗
− λ 1 I 2 )u 1 = 0 ,
(J
∗ J − λ 1 I 2 )v 1 = 0 ,
(JJ
∗
− λ 2 I 2 )u 2 = 0 ,
(J
∗ J − λ 2 I 2 )v 2 = 0 ,
(1.37)
or
−24.329 317
9
9
−3.329 317
u 11
u 21
= 0 ,
−27.329 317
3
3
−0.329 317
v 11
v 21
= 0 ,
3.329 317
9
9
24.329 317
u 12
u 22
= 0 ,
0.329 317
3
3
27.329 317
v 12
v 22
= 0 .
(1.38)
Note that the matrices JJ
∗
− λI 2 and J
∗ J − λI 2 have only rank one. Accordingly, in order to solve the
homogenous linear equations uniquely, we need an additional constraint. Conventionally, this problem
is solved by postulating normalized eigencolumns, namely
u
2
11 + u
2
21 = 1 , u
2
12 + u
2
22 = 1 , v
2
11 + v
2
21 = 1 , v
2
12 + v
2
22 = 1 ,
u 1 = u 2 = 1 ,
v 1 = v 2 = 1 .
(1.39)
The left eigencolumns, which are here denoted as (u 1 , u 2 ), are constructed from the following system
of equations:
−24.329 317u 11 + 9u 21 = 0 , +3.329 317u 12 + 9u 22 = 0 ,
u
2
11 + u
2
21 = 1 , u
2
12 + u
2
22 = 1 .
(1.40)