112 2 From Riemann manifolds to Euclidean manifolds
F l = R 1 S 1 versus F r = R 3 S 3
versus
versus
F l = S 2 R 2 versus F r = S 4 R 4
,
(2.69)
where the matrices R i are orthonormal, R
−1
i
= R
T
i , while the matrices S i are by definition symmetric,
S i = S
T
i . These symmetric matrices S i are sometimes called stretch matrices. For more details including
numerical examples, we refer to J. E. Marsden and T. J. R. Hughes (1983, pp. 51–55), R. W. Ogden
(1984, pp. 92–94), J. C. Simo and R. L. Taylor (1991), and T. C. T. Ting (1985). Here, we conclude with
a second remark relating again to the simultaneous diagonalization of two matrices, e.,g. the pairs of
Cauchy-Green deformation tensors {C l , G l } or {C r , G r } and the pairs of Euler-Lagrange deformation
tensors {E l , G l } or {E r , G r }, respectively. Of course, we could also aim at a simultaneous diagonalization
of three matrices, e. g. the triplets
{E l , C l , G l } versus {E r , C r , G r } ,
(2.70)
in particular
U
T
l G l X l = S
1
l ⇔ G l = U l S
1
l X
−1
l
versus G r = U r S
1
r X
−1
r ⇔ U
T
r G r X r = S
1
r ,
(2.71)
X
T
l C l Y l = S
2
l ⇔ C l =
X
−1
l
T S
2
l Y
−1
l
versus C r =
X
−1
r
T S
2
r Y
−1
r ⇔ X
T
r C r Y r = S
2
r , (2.72)
Y
T
l E l V l = S
3
l ⇔ E l =
Y
−1
l
T S
3
l V
T
l
versus E r =
Y
−1
r
T S
3
r V
T
r ⇔ Y
T
r E r V r = S
3
r ,
(2.73)
where S
1 , S
2 , and S
3 are certain quasi-diagonal matrices, where V and U are unitary matrices, and
non-singular matrices are X l , Y l and X r , Y r , respectively. But we are not able to diagonalize G l and
G r , respectively, to unity. The diagonalization of G l and G r , respectively, to unit matrices is by all
means recommendable since accordingly all other tensors, e.g. C l and C r , respectively, or E l and E r ,
alternatively, refer to unit vectors which span the local tangent space of M
2
l or M
2
r , respectively.
Before we proceed to the next chapter, let us here additionally note that a tree of generalization of
the ordinary singular value decompositions has been developed by M. T. Chu (1991 a,b), B. de Moor
and H. Zha (1991), H. Zha (1991), and others to which we refer.
F l = R 1 S 1 versus F r = R 3 S 3
versus
versus
F l = S 2 R 2 versus F r = S 4 R 4
,
(2.69)
where the matrices R i are orthonormal, R
−1
i
= R
T
i , while the matrices S i are by definition symmetric,
S i = S
T
i . These symmetric matrices S i are sometimes called stretch matrices. For more details including
numerical examples, we refer to J. E. Marsden and T. J. R. Hughes (1983, pp. 51–55), R. W. Ogden
(1984, pp. 92–94), J. C. Simo and R. L. Taylor (1991), and T. C. T. Ting (1985). Here, we conclude with
a second remark relating again to the simultaneous diagonalization of two matrices, e.,g. the pairs of
Cauchy-Green deformation tensors {C l , G l } or {C r , G r } and the pairs of Euler-Lagrange deformation
tensors {E l , G l } or {E r , G r }, respectively. Of course, we could also aim at a simultaneous diagonalization
of three matrices, e. g. the triplets
{E l , C l , G l } versus {E r , C r , G r } ,
(2.70)
in particular
U
T
l G l X l = S
1
l ⇔ G l = U l S
1
l X
−1
l
versus G r = U r S
1
r X
−1
r ⇔ U
T
r G r X r = S
1
r ,
(2.71)
X
T
l C l Y l = S
2
l ⇔ C l =
X
−1
l
T S
2
l Y
−1
l
versus C r =
X
−1
r
T S
2
r Y
−1
r ⇔ X
T
r C r Y r = S
2
r , (2.72)
Y
T
l E l V l = S
3
l ⇔ E l =
Y
−1
l
T S
3
l V
T
l
versus E r =
Y
−1
r
T S
3
r V
T
r ⇔ Y
T
r E r V r = S
3
r ,
(2.73)
where S
1 , S
2 , and S
3 are certain quasi-diagonal matrices, where V and U are unitary matrices, and
non-singular matrices are X l , Y l and X r , Y r , respectively. But we are not able to diagonalize G l and
G r , respectively, to unity. The diagonalization of G l and G r , respectively, to unit matrices is by all
means recommendable since accordingly all other tensors, e.g. C l and C r , respectively, or E l and E r ,
alternatively, refer to unit vectors which span the local tangent space of M
2
l or M
2
r , respectively.
Before we proceed to the next chapter, let us here additionally note that a tree of generalization of
the ordinary singular value decompositions has been developed by M. T. Chu (1991 a,b), B. de Moor
and H. Zha (1991), H. Zha (1991), and others to which we refer.