From Riemann Manifolds to Euclidean Manifolds - Map Projections: Cartographic Information Systems - page 111

a right Euclidean manifold. Second, we focus on an interpretation of the results and additionally
discuss a short example.

Lemma 2.1 (Left and right eigenspace analysis and synthesis of the Cauchy-Green deformation
tensor, special case

r ,g μν 2 ,δ μν

{ M } = { R } ).

(i) Synthesis.

For the matrix pair of positive-definite and symmetric matrices { C l , G l } or { C r , G r } , a simulta-

neous diagonalization is (the right Frobenius matrix F r is an orthonormal matrix)

C l =J l J l , F l C l F l =diag[ Λ 1 ,Λ 2 ] , F l G l F l
=IversusF r C r F r =diag[ λ 1 ,λ 2 ] , F r F r =I . (2.1)

(ii) Analysis.

Left eigenvalues or left principal stretches:

| C l − Λ i G l | =0 ,

tr[C l G − l ]

4det[C l G − l ] .

(tr[C l G − l ]) 2

= 1
2

Λ 1 , 2 = Λ 2

± − (2.2)

±

Left eigencolumns:

F 11
F 21

=

1

G 11 ( c 22 − Λ 1 G 22 ) 2

− 2 G 12 ( c 12 − Λ 1 G 12 )( c 22 − Λ 1 G 22 )+ G 22 ( c 12 − Λ 1 G 12 ) 2 ×

+( c 22 − ,

Λ 1 G 22 )

× (2.3)

Λ 1 G 12 )

− ( c 12 −

F 12
F 22

=

1

G 22 ( c 11 −

Λ 2 G 12 ) 2 × Λ 2 G 11 ) 2 − 2 G 12 ( c 11 − Λ 2 G 11 )( c 12 − Λ 2 G 12 )+ G 11 ( c 12 −

− .

Λ 2 G 12 )
+( c 11 − Λ 2 G 11 )

( c 12 −

×

Right eigenvalues or right principal stretches

(the right general eigenvalue problem reduces to the right special eigenvalue problem):

| C r − λ i G r | = | C r − λ i I 2 | =0 ∀ i ∈{ 1 , 2 },

tr[C r G − r ] ±

− 4det [C r G − r ] =

tr[C r G − r ]) 2

= 1
2

λ 1 , 2 = λ 2

(2.4)

±

C 11 + C 22 ±

( C 11 + C 22 ) 2 (2 C 12 ) 2 .

= 1
2