In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number z as
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Matrix polar decomposition
The polar decomposition of a square complex matrix A is a matrix decomposition of the form
where U is a unitary matrix and P is a positive-semidefinite Hermitian matrix. Intuitively, the polar decomposition separates A into a component that stretches the space along a set of orthogonal axes, represented by P, and a rotation (with possible reflection) represented by U. The decomposition of the complex conjugate of
This decomposition always exists; and so long as A is invertible, it is unique, with P positive-definite. Note that
gives the corresponding polar decomposition of the determinant of A, since
The matrix P is always unique, even if A is singular, and given by
where A* denotes the conjugate transpose of A. This expression is meaningful since a positive-semidefinite Hermitian matrix has a unique positive-semidefinite square root. If A is invertible, then the matrix U is given by
In terms of the singular value decomposition of A, A = W Σ V*, one has
confirming that P is positive-definite and U is unitary. Thus, the existence of the SVD is equivalent to the existence of polar decomposition.
One can also decompose A in the form
Here U is the same as before and P′ 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 A is normal if and only if P′ = P. Then UΣ = ΣU, and it is possible to diagonalise U with a unitary similarity matrix S that commutes with Σ, giving S U S* = Φ−1, where Φ is a diagonal unitary matrix of phases eiφ. Putting Q = V S*, one can then re-write the polar decomposition as
so A then thus also has a spectral decomposition
with complex eigenvalues such that ΛΛ* = Σ2 and a unitary matrix of complex eigenvectors Q.
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(N), then |A| = {A*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:
Lemma If A, B are bounded operators on a Hilbert space H, and A*A ≤ B*B, then there exists a contraction C such that A = CB. Furthermore, C is unique if Ker(B*) ⊂ Ker(C).The operator C can be defined by C(Bh) := Ah for all h in H, extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement to all of H. The lemma then follows since A*A ≤ B*B implies Ker(A) ⊂ Ker(B).
In particular. If A*A = B*B, then C is a partial isometry, which is unique if Ker(B*) ⊂ Ker(C). In general, for any bounded operator A,
where (A*A)½ is the unique positive square root of A*A given by the usual functional calculus. So by the lemma, we have
for some partial isometry U, which is unique if Ker(A*) ⊂ Ker(U). Take P to be (A*A)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 (see example above). 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 (unique) polar decomposition
where |A| is a (possibly unbounded) 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(|A|).
The proof uses the same lemma as above, which goes through for unbounded operators in general. If Dom(A*A) = Dom(B*B) and A*Ah = B*Bh for all h ∈ Dom(A*A), then there exists a partial isometry U such that A = UB. U is unique if Ran(B)⊥⊂ Ker(U). The operator A being closed and densely defined ensures that the operator A*A is self-adjoint (with dense domain) and therefore allows one to define (A*A)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(P), for any Borel set B in [0, ∞).
Quaternion polar decomposition
The polar decomposition of quaternions H depends on the sphere
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 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: