Polar decomposition


In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form, where is a unitary matrix and is a positive-semidefinite Hermitian matrix, both square and of the same size.
Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space, the polar decomposition separates it into a rotation or reflection of, and a scaling of the space along a set of orthogonal axes.
The polar decomposition of a square matrix always exists. If is invertible, the decomposition is unique, and the factor will be positive-definite. In that case, can be written uniquely in the form, where is unitary and is the unique self-adjoint logarithm of the matrix. This decomposition is useful in computing the fundamental group of Lie groups.
The polar decomposition can also be defined as where and have the same properties as above.
The polar decomposition of a matrix can be seen as the matrix analog of the polar form of a complex number as, where is its absolute value, and is a complex number with unit norm.

Properties

The polar decomposition of the complex conjugate of is given by Note that
gives the corresponding polar decomposition of the determinant of A, since and. In particular, if has determinant 1 then both and have determinant 1.
The positive-semidefinite matrix P is always unique, even if A is singular, and is denoted as
where A denotes the conjugate transpose of A. The uniqueness of P ensures that this expression is well-defined. The uniqueness is guaranteed by the fact that is a positive-semidefinite Hermitian matrix and, therefore, has a unique positive-semidefinite Hermitian square root. If A is invertible, then P is positive-definite, thus also invertible and the matrix U is uniquely determined by

Intuitive interpretation

A real square matrix can be interepreted as the linear transformation of that takes a column vector to. Then, in the polar decomposition, the factor is an real orthonormal matrix. The polar decomposition then can be seen as expressing the linear transformation defined by into a scaling of the space along each eigenvector of by a scale factor , followed by a single rotation or reflection of .
Alternatively, the decomposition expresses the transformation defined by as a rotation followed by a scaling along certain orthogonal directions. The scale factors are the same, but the directions are different.

Relation to the SVD

In terms of the singular value decomposition of,, one has
where,, and are unitary matrices. This confirms that is positive-definite and is unitary. Thus, the existence of the SVD is equivalent to the existence of polar decomposition.
One can also decompose in the form
Here is the same as before and is given by
This is known as the left polar decomposition, whereas the previous decomposition is known as the right polar decomposition. Left polar decomposition is also known as reverse polar decomposition.
The matrix is normal if and only if. Then, and it is possible to diagonalise with a unitary similarity matrix that commutes with, giving, where is a diagonal unitary matrix of phases. Putting, one can then re-write the polar decomposition as
so then thus also has a spectral decomposition
with complex eigenvalues such that and a unitary matrix of complex eigenvectors.
The polar decomposition of a square invertible real matrix is of the form
where is a positive-definite matrix and is an orthogonal matrix.

Construction and proofs of existence

The core idea behind the construction of the polar decomposition is similar to that used to compute the singular-value decomposition.
For any, the matrix is Hermitian and positive semi-definite, and therefore unitarily equivalent to a positive semi-definite diagonal matrix. Let then be the unitary such that, with diagonal and positive semi-definite.

Case of A normal

If is normal, then it is unitarily equivalent to a diagonal matrix: for some unitary and some diagonal matrix. We can then writewhere is a diagonal matrix with containing the phases of the elements of, that is, or an arbitrary complex number with unit magnitude when.
The polar decomposition is thus, with and diagonal in the eigenbasis of and with eigenvalues equal to the phases and absolute values of the eigenvalues of, respectively.

Case of A invertible

From the singular-value decomposition, it can be shown that a is invertible if and only if is. Moreover, this is true if and only if the eigenvalues of are all not zero.
In this case, the polar decomposition is directly obtained by writing
and observing that is unitary. To see this, we can exploit the spectral decomposition of to write.
In this expression, is unitary because is. To show that also is unitary, we can use the SVD to write, so that
where again is unitary by construction.
Yet another way to directly show the unitarity of is to note that, writing the SVD of in terms of rank-1 matrices as, where are the singular values of, we have which directly implies the unitarity of because a matrix is unitary if and only if its singular values have unitary absolute value.
Note how, from the above construction, it follows that the unitary matrix in the polar decomposition of an invertible matrix is uniquely defined.

General case

The SVD of reads, with unitary matrices, and a diagonal, positive semi-definite matrix. By simply inserting an additional pair of s or s, we obtain the two forms of the polar decomposition of :

Bounded operators on Hilbert space

The polar decomposition of any bounded linear operator A between complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator.
The polar decomposition for matrices generalizes as follows: if A is a bounded linear operator then there is a unique factorization of A as a product A = UP where U is a partial isometry, P is a non-negative self-adjoint operator and the initial space of U is the closure of the range of P.
The operator U must be weakened to a partial isometry, rather than unitary, because of the following issues. If A is the one-sided shift on l2, then |A| = ½ = I. So if A = U |A|, U must be A, which is not unitary.
The existence of a polar decomposition is a consequence of Douglas' lemma:
The operator C can be defined by C := Ah for all h in H, extended by continuity to the closure of Ran, and by zero on the orthogonal complement to all of H. The lemma then follows since AA ≤ BB implies Ker ⊂ Ker.
In particular. If AA = BB, then C is a partial isometry, which is unique if Ker ⊂ Ker.
In general, for any bounded operator A,
where ½ is the unique positive square root of AA given by the usual functional calculus. So by the lemma, we have
for some partial isometry U, which is unique if Ker ⊂ Ker. Take P to be ½ and one obtains the polar decomposition A = UP. Notice that an analogous argument can be used to show A = P'U, where P' is positive and U a partial isometry.
When H is finite-dimensional, U can be extended to a unitary operator; this is not true in general. Alternatively, the polar decomposition can be shown using the operator version of singular value decomposition.
By property of the continuous functional calculus, |A| is in the C*-algebra generated by A. A similar but weaker statement holds for the partial isometry: U is in the von Neumann algebra generated by A. If A is invertible, the polar part U will be in the C*-algebra as well.

Unbounded operators

If A is a closed, densely defined unbounded operator between complex Hilbert spaces then it still has a polar decomposition
where |A| is a non-negative self adjoint operator with the same domain as A, and U is a partial isometry vanishing on the orthogonal complement of the range Ran.
The proof uses the same lemma as above, which goes through for unbounded operators in general. If Dom = Dom and AAh = BBh for all h ∈ Dom, then there exists a partial isometry U such that A = UB. U is unique if Ran⊥ ⊂ Ker. The operator A being closed and densely defined ensures that the operator AA is self-adjoint and therefore allows one to define ½. Applying the lemma gives polar decomposition.
If an unbounded operator A is affiliated to a von Neumann algebra M, and A = UP is its polar decomposition, then U is in M and so is the spectral projection of P, 1B.

Quaternion polar decomposition

The polar decomposition of quaternions H depends on the unit 2-dimensional sphere of square roots of minus one. Given any r on this sphere, and an angle −π < a ≤ π, the versor is on the unit 3-sphere of H. For a = 0 and a = π, the versor is 1 or −1 regardless of which r is selected. The norm t of a quaternion q is the Euclidean distance from the origin to q. When a quaternion is not just a real number, then there is a unique polar decomposition

Alternative planar decompositions

In the Cartesian plane, alternative planar ring decompositions arise as follows:

Numerical determination of the matrix polar decomposition

To compute an approximation of the polar decomposition A = UP, usually the unitary factor U is approximated. The iteration is based on Heron's method for the square root of 1 and computes, starting from, the sequence
The combination of inversion and Hermite conjugation is chosen so that in the singular value decomposition, the unitary factors remain the same and the iteration reduces to Heron's method on the singular values.
This basic iteration may be refined to speed up the process: