8
1 From Riemann manifolds to Riemann manifolds
There exists an intriguing representation of the matrix of deformation gradients J as well as of
the matrix of Cauchy–Green deformation C, namely the polar decomposition. It is a generalization to
matrices of the familiar polar representation of a complex number z = r exp iφ, (r ≥ 0) and is defined
in Corollary 1.3.
Corollary 1.3 (Polar decomposition).
Let J ∈ R
n×n . Then there exists a unique orthonormal matrix R ∈ SO(n) (called rotation matrix)
and a unique symmetric positive-definite matrix S (called stretch) such that (1.26) holds and the
expressions (1.27) are a polar decomposition of the matrix of Cauchy–Green deformation.
J = RS , R
∗ R = I n , S = S
∗ ,
(1.26)
C l = J
∗
l G r J l = S l R
∗ G r RS l
versus
S r R
∗ G l RS r = J
∗
r G l J r = C r .
(1.27)
End of Corollary.
Question.
Question: “How can we compute the polar decomposition of the Jacobi matrix?” Answer:
“An elegant way is the singular value decomposition defined in Corollary 1.4.”
Corollary 1.4 (Polar decomposition by singular value decomposition).
Let the matrix J ∈ R
2×2 have the singular value decomposition J = UΣV
∗ , where the matrices U ∈ R
2×2
and V ∈ R
2×2 are orthonormal (unitary), i. e. U
∗ U = I 2 and V
∗ V = I 2 , and where Σ = diag(σ 1 , σ 2 ) in
descending order σ 1 ≥ σ 2 ≥ 0 is the diagonal matrix of singular values {σ 1 , σ 2 }. If J has the polar
decomposition J = RS, then R = UV
∗ and S = VΣV
∗ . λ(J) and σ(J) denote, respectively, the set of
eigenvalues and the set of singular values of J. Then
the left eigenspace is spanned by the left eigencolumns u 1 and u 2 which are generated by
(JJ
∗
− λ i I 2 )u i = (JJ
∗
− σ
2
i I 2 )u i = 0 , ||u 1 || = ||u 2 || = 1 ;
(1.28)
the right eigenspace is spanned by the right eigencolumns v 1 and v 2 generated by
(J
∗ J − λ j I 2 )v j = (J
∗ J − σ
2
j I 2 )v j = 0 , ||v 1 || = ||v 2 || = 1 ;
(1.29)
the characteristic equation of the eigenvalues is determined by
|JJ
∗
− λI 2 | = 0 or |J
∗ J − λI 2 | = 0 ,
(1.30)
which leads to λ
2
− λI + II = 0, with the invariants
I := tr [JJ
∗ ] = tr [J
∗ J] , II := (det [J])
2 = det [JJ
∗ ] = det [J
∗ J] ,
λ 1 = σ
2
1 =
1
2
I +
√
I 2 − 4II
, λ 2 = σ
2
2 =
1
2
I −
√
I 2 − 4II
;
(1.31)
the matrices S and R can be expressed as
S = (J
∗ J)
1/2 = (v 1 , v 2 )diag(σ 1 , σ 2 )(v
∗
1 , v
∗
2 ) , R = JS
−1 = (u 1 , u 2 )(v
∗
1 , v
∗
2 ) ;
(1.32)
J is normal if and only if RS = SR.
End of Corollary.
1 From Riemann manifolds to Riemann manifolds
There exists an intriguing representation of the matrix of deformation gradients J as well as of
the matrix of Cauchy–Green deformation C, namely the polar decomposition. It is a generalization to
matrices of the familiar polar representation of a complex number z = r exp iφ, (r ≥ 0) and is defined
in Corollary 1.3.
Corollary 1.3 (Polar decomposition).
Let J ∈ R
n×n . Then there exists a unique orthonormal matrix R ∈ SO(n) (called rotation matrix)
and a unique symmetric positive-definite matrix S (called stretch) such that (1.26) holds and the
expressions (1.27) are a polar decomposition of the matrix of Cauchy–Green deformation.
J = RS , R
∗ R = I n , S = S
∗ ,
(1.26)
C l = J
∗
l G r J l = S l R
∗ G r RS l
versus
S r R
∗ G l RS r = J
∗
r G l J r = C r .
(1.27)
End of Corollary.
Question.
Question: “How can we compute the polar decomposition of the Jacobi matrix?” Answer:
“An elegant way is the singular value decomposition defined in Corollary 1.4.”
Corollary 1.4 (Polar decomposition by singular value decomposition).
Let the matrix J ∈ R
2×2 have the singular value decomposition J = UΣV
∗ , where the matrices U ∈ R
2×2
and V ∈ R
2×2 are orthonormal (unitary), i. e. U
∗ U = I 2 and V
∗ V = I 2 , and where Σ = diag(σ 1 , σ 2 ) in
descending order σ 1 ≥ σ 2 ≥ 0 is the diagonal matrix of singular values {σ 1 , σ 2 }. If J has the polar
decomposition J = RS, then R = UV
∗ and S = VΣV
∗ . λ(J) and σ(J) denote, respectively, the set of
eigenvalues and the set of singular values of J. Then
the left eigenspace is spanned by the left eigencolumns u 1 and u 2 which are generated by
(JJ
∗
− λ i I 2 )u i = (JJ
∗
− σ
2
i I 2 )u i = 0 , ||u 1 || = ||u 2 || = 1 ;
(1.28)
the right eigenspace is spanned by the right eigencolumns v 1 and v 2 generated by
(J
∗ J − λ j I 2 )v j = (J
∗ J − σ
2
j I 2 )v j = 0 , ||v 1 || = ||v 2 || = 1 ;
(1.29)
the characteristic equation of the eigenvalues is determined by
|JJ
∗
− λI 2 | = 0 or |J
∗ J − λI 2 | = 0 ,
(1.30)
which leads to λ
2
− λI + II = 0, with the invariants
I := tr [JJ
∗ ] = tr [J
∗ J] , II := (det [J])
2 = det [JJ
∗ ] = det [J
∗ J] ,
λ 1 = σ
2
1 =
1
2
I +
√
I 2 − 4II
, λ 2 = σ
2
2 =
1
2
I −
√
I 2 − 4II
;
(1.31)
the matrices S and R can be expressed as
S = (J
∗ J)
1/2 = (v 1 , v 2 )diag(σ 1 , σ 2 )(v
∗
1 , v
∗
2 ) , R = JS
−1 = (u 1 , u 2 )(v
∗
1 , v
∗
2 ) ;
(1.32)
J is normal if and only if RS = SR.
End of Corollary.