A.5 The Right Polar Factorization
The right polar factorization involves writing a matrix as a product of two other matrices,
one which preserves distances and the other which stretches and distorts. First
here are some lemmas which review and add to many of the topics discussed so
far about adjoints and orthonormal sets and such things. This is of fundamental
significance in geometric measure theory and also in continuum mechanics. Not
surprisingly the stress should depend on the part which stretches and distorts. See
Lemma A.5.1 Let A be a Hermitian matrix such that all its eigenvalues are
nonnegative. Then there exists a Hermitian matrix A1∕2 such that A1∕2 has all
nonnegative eigenvalues and
Proof: Since A is Hermitian, there exists a diagonal matrix D having all real nonnegative
entries and a unitary matrix U such that A = U∗DU. This is from Theorem A.4.1 above.
Then denote by D1∕2 the matrix which is obtained by replacing each diagonal entry of D with
its square root. Thus D1∕2D1∕2 = D. Then define
Since D1∕2 is real,
so A1∕2 is Hermitian. ■
Next it is helpful to recall the Gram Schmidt algorithm and observe a certain property
stated in the next lemma.
Lemma A.5.2 Suppose
is a linearly independent set of
vectors such that
is an orthonormal set of vectors. Then when the Gram
Schmidt process is applied to the vectors in the given order, it will not change any of
be the orthonormal set delivered by the Gram Schmidt process.
because by definition, u1 ≡ w1∕
Now suppose uj
j ≤ k ≤ r.
Then if k < r,
consider the definition of uk+1.
By induction, uj = wj and so this reduces to wk+1∕
This lemma immediately implies the following lemma.
Lemma A.5.3 Let V be a subspace of dimension p and let
orthonormal set of vectors in V . Then this orthonormal set of vectors may be extended to an
orthonormal basis for V,
Proof: First extend the given linearly independent set
to a basis for
then apply the Gram Schmidt theorem to the resulting basis. Since
orthonormal it follows from Lemma
the result is of the desired form, an orthonormal
Here is another lemma about preserving distance.
Lemma A.5.4 Suppose R is an m×n matrix with m > n and R preserves distances.
Then R∗R = I. Also, if R takes an orthonormal basis to an orthonormal set, then R
must preserve distances.
Proof: Since R preserves distances,
Therefore from the axioms of
the dot product,
and so for all x,y,
Hence for all x,y,
Now for a x,y given, choose α ∈ ℂ such that
= 0 for all
because the given x,y
were arbitrary. Let
= R∗Rx − x
to conclude that for all x,
which says R∗R = I since x is arbitrary.
Consider the last claim. Let R : Fn → Fm such that
is an orthonormal basis
is also an orthormal set, then
With this preparation, here is the big theorem about the right polar factorization.
Theorem A.5.5 Let F be an m × n matrix where m ≥ n. Then there exists a
Hermitian n × n matrix U which has all nonnegative eigenvalues and an m × n matrix R
which satisfies R∗R = I such that
Proof: Consider F∗F. This is a Hermitian matrix because
Also the eigenvalues of the n × n matrix F∗F are all nonnegative. This is because if x is an
Therefore, by Lemma A.5.1, there exists an n×n Hermitian matrix U having all nonnegative
eigenvalues such that
Consider the subspace U
be an orthonormal basis for
Note that U
might not be all of
Using Lemma A.5.3
, extend to an orthonormal basis
for all of Fn,
Next observe that
is also an orthonormal set of vectors in
. This is
Therefore, from Lemma A.5.3
again, this orthonormal set of vectors can be extended to an
orthonormal basis for Fm,
Thus there are at least as many zk as there are yj. Now for x ∈ Fn, since
is an orthonormal basis for Fn, there exist unique scalars,
Then also there exist scalars bk such that
and so from 1.7,
and this shows
it follows that R
maps an orthonormal set to an orthonormal set and so R
preserves distances. Therefore, by
Lemma A.5.4 R∗R