An application of Theorem 12.3.2, is the following fundamental result, important in geometric
measure theory and continuum mechanics. It is sometimes called the right polar decomposition.
The notation used is that which is seen in continuum mechanics, see for example Gurtin [12].
Don’t confuse the U in this theorem with a unitary transformation. It is not so. When the
following theorem is applied in continuum mechanics, F is normally the deformation gradient, the
derivative of a nonlinear map from some subset of three dimensional space to three
dimensional space. In this context, U is called the right Cauchy Green strain tensor. It is a
measure of how a body is stretched independent of rigid motions. First, here is a simple
lemma.
Lemma 12.7.1Suppose R ∈ ℒ
(X,Y )
where X,Y are inner product spaces and Rpreserves distances. Then R^{∗}R = I.
Proof:Since R preserves distances,
|Ru |
=
|u |
for every u. Let u,v be arbitrary vectors in X
and let θ ∈ ℂ,
( ( ∑r ) (∑r ) )
F bkxk − F (x),F bkxk − F (x)
k=1 k=1
( (∑r ) (∑r ) )
= (F ∗F ) bkxk − x , bkxk − x
k=1 k=1
( (∑r ) (∑r ) )
= U 2 bkxk − x , bkxk − x
k=1 k=1
( ( ∑r ) (∑r ) )
= U bkxk − x ,U bkxk − x
( k=1 k=1 )
∑r ∑r
= bkU xk − U x, bkU xk − U x = 0
k=1 k=1
Because from 12.14, Ux = ∑_{k=1}^{r}b_{k}Ux_{k}. Therefore, RUx = F
∑r
( k=1bkxk)
= F
(x)
.
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The following corollary follows as a simple consequence of this theorem. It is called the left
polar decomposition.
Corollary 12.7.3Let F ∈ℒ
(X, Y)
and suppose n ≥ m where X is a inner product space ofdimension n and Y is a inner product space of dimension m. Then there exists a Hermitian U ∈ℒ
(X,X )
, and an element of ℒ
(X, Y)
,R, such that
F = UR, RR ∗ = I.
Proof:Recall that L^{∗∗} = L and
(M L)
^{∗} = L^{∗}M^{∗}. Now apply Theorem 12.7.2 to
F^{∗}∈ℒ
(Y,X )
. Thus, F^{∗} = R^{∗}U where R^{∗} and U satisfy the conditions of that theorem. Then
F = UR and RR^{∗} = R^{∗∗}R^{∗} = I. ■
The following existence theorem for the polar decomposition of an element of ℒ
(X, X )
is a
corollary.
Corollary 12.7.4Let F ∈ℒ
(X,X )
. Then there exists a HermitianW ∈ℒ
(X, X )
, anda unitary matrix Q such that F = WQ, and there exists a Hermitian U ∈ℒ
(X,X )
and aunitary R, such that F = RU.
This corollary has a fascinating relation to the question whether a given linear transformation
is normal. Recall that an n×n matrix A, is normal if AA^{∗} = A^{∗}A. Retain the same definition for
an element of ℒ
(X, X )
.
Theorem 12.7.5Let F ∈ℒ
(X, X)
. Then F is normal if and only if in Corollary 12.7.4RU = UR and QW = WQ.
Proof: I will prove the statement about RU = UR and leave the other part as an exercise.
First suppose that RU = UR and show F is normal. To begin with,
∗ ∗ ∗ ∗
U R = (RU ) = (U R) = R U.
Therefore,
∗ ∗ 2
F F = UR RU = U
FF ∗ = RU UR ∗ = U RR∗U = U 2
which shows F is normal.
Now suppose F is normal. Is RU = UR? Since F is normal,
∗ ∗ 2 ∗
FF = RU U R = RU R
and
F ∗F = UR ∗RU = U 2.
Therefore, RU^{2}R^{∗} = U^{2}, and both are nonnegative and self adjoint. Therefore, the
square roots of both sides must be equal by the uniqueness part of the theorem on
fractional powers. It follows that the square root of the first, RUR^{∗} must equal the
square root of the second, U. Therefore, RUR^{∗} = U and so RU = UR. This proves the
theorem in one case. The other case in which W and Q commute is left as an exercise.
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