This Sectionisa Draft, Subjectto Changes Needs Numerical Examples
The polar decomposition of a matrix writes any matrix as the product
of a unitary matrix (Definition UM)and a positive semi-definite matrix
(Definition PSM). It takes its name from a special way to write complex numbers.
If you’ve had a basic course in complex analysis, the next paragraph will help
explain the name. If the next paragraph makes no sense to you, there’s no harm in
skipping it.
Any complex number z∈ℂ
can be written as z=reiθ
where r is a
positive number (computed as a square root of a function of the real amd imaginary parts
of z) and
θ is an angle of rotation
that converts 1 to the
complex number eiθ=cos(θ)+isin(θ).
The polar form of a square matrix is a product of a positive semi-definite matrix
that is a square root of a function of the matrix together with a unitary matrix,
which can be viewed as achieving a rotation (Theorem UMPIP).
OK, enough preliminaries. We have all the tools in place to jump straight to
our main theorem.
Theorem PDM Polar Decomposition of a Matrix Suppose that A
is a square matrix. Then there is a unitary matrix
U such
that A=AA∗1∕2U.
□
Proof This theorem only claims the existence of a unitary matrix
U that does a certain job.
We will manufacture U
and check that it meets the requirements.
Suppose A has
size n and rank
r. We begin by applying
Theorem EEMAP to A.
Let B=x1x2x3…xn be the orthonormal
basis of ℂn composed of
eigenvectors for A∗A, and
let C=y1y2y3…yn be the orthonormal
basis of ℂn composed
of eigenvectors for AA∗.
We have Axi=δixi,
1≤i≤r, and
Axi=0,
r+1≤i≤n, where
δi,
1≤i≤r are the distinct
nonzero eigenvalues of A∗A.
Define T:ℂn→ℂn
to be the unique linear transformation such that
Txi=yi,
1≤i≤n, as guaranteed by
Theorem LTDB. Let E
be the basis of standard unit vectors for
ℂn (Definition SUV),
and define U
to be the matrix representation (Definition MR) of
T with respect
to E, more
carefully U=MTEE.
This is the matrix we are after. Notice that
Since B and
C are orthonormal bases,
and C is the result of
multiplying the vectors of B
by U, we
conclude that U
is unitary by Theorem UMCOB. So once again, Theorem EEMAP is a big part
of the setup for a decomposition.
Let x∈ℂn be any
vector. Since B is a
basis of ℂn, there are
scalars a1a2a3…an expressing
x as a linear combination
of the vectors in B.
then