From Riemann Manifolds to Euclidean Manifolds - Map Projections: Cartographic Information Systems - page 127

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

{ . respectively. Of course, we could also aim at a simultaneous diagonalization

of three matrices, e.g. the triplets

E l , G l } or { E r , G r }

{ E l , C l , G l } versus { E r , C r , G r } , (2.70)

in particular

U l G l X l =S l ⇔ G l =U l S l X − 1 versus G r =U r S r X − 1

⇔ U r G r X r =S r ,

(2.71)

i r

X l C l Y l =S l ⇔ C l =(X − l ) T S l Y − l versus C r =(X − r ) T S r Y − 1 X r C r Y r =S r ,

⇔ (2.72)

r

Y l E l V l =S l ⇔ E l =(Y − l ) T S l V l versus E r =(Y − r ) T S r V r ⇔ Y r E r V r =S r , (2.73)

where S 1 , S 2 ,andS 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

l or

M

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 Chu
( 1991a , b ), De Moor and Zha ( 1991 ), Zha ( 1991 ), and others to which we refer.