Polar decomposition

In mathematics, the polar decomposition of a square real or complex matrix formula_1 is a factorization of the form formula_2, where formula_3 is a unitary matrix and formula_4 is a positive-semidefinite Hermitian matrix, both square and of the same size. 
Intuitively, if a real formula_5 matrix formula_1 is interpreted as a linear transformation of formula_7-dimensional space formula_8, the polar decomposition separates it into a rotation or reflection formula_3 of formula_8, and a scaling of the space along a set of formula_7 orthogonal axes. 
The polar decomposition of a square matrix formula_1 always exists. If formula_1 is invertible, the decomposition is unique, and the factor formula_4 will be positive-definite. In that case, formula_1 can be written uniquely in the form formula_16, where formula_3 is unitary and formula_18 is the unique self-adjoint logarithm of the matrix formula_4. This decomposition is useful in computing the fundamental group of (matrix) Lie groups.
The polar decomposition can also be defined as formula_20 where formula_4 is symmetric positive-definite but is in general a different matrix, while formula_3 is the same matrix as above.
The polar decomposition of a matrix can be seen as the matrix analog of the polar form of a complex number formula_23 as formula_24, where formula_25 is its absolute value (a non-negative real number), and formula_26 is a complex number with unit norm (an element of the circle group).
The definition formula_27 may be extended to rectangular matrices formula_28 by requiring formula_29 to be a semi-unitary matrix and formula_30 to be a positive-semidefinite Hermitian matrix. The decomposition always exists and formula_4 is always unique. The matrix formula_3 is unique if and only if formula_1 has full rank. 
Properties.
The polar decomposition of the complex conjugate of formula_1 is given by formula_35 Note that
gives the corresponding polar decomposition of the determinant of "A", since formula_37 and formula_38. In particular, if formula_1 has determinant 1 then both formula_3 and formula_4 have determinant 1.
The positive-semidefinite matrix "P" is always unique, even if "A" is singular, and is denoted as
where formula_43 denotes the conjugate transpose of formula_1. The uniqueness of "P" ensures that this expression is well-defined. The uniqueness is guaranteed by the fact that formula_45 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 formula_47 matrix formula_1 can be interepreted as the linear transformation of formula_49 that takes a column vector formula_50 to formula_51. Then, in the polar decomposition formula_52, the factor formula_53 is an formula_47 real orthonormal matrix. The polar decomposition then can be seen as expressing the linear transformation defined by formula_1 into a scaling of the space formula_49 along each eigenvector formula_57 of formula_1 by a scale factor formula_59 (the action of formula_4), followed by a single rotation or reflection of formula_49 (the action of formula_53). 
Alternatively, the decomposition formula_63 expresses the transformation defined by formula_1 as a rotation (formula_53) followed by a scaling (formula_4) 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 (SVD) of formula_1, formula_68, one has
where formula_3, formula_71, and formula_72 are unitary matrices (called orthogonal matrices if the field is the reals formula_73). This confirms that formula_4 is positive-definite and formula_3 is unitary. Thus, the existence of the SVD is equivalent to the existence of polar decomposition.
One can also decompose formula_1 in the form
Here formula_3 is the same as before and formula_79 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 formula_1 is normal if and only if formula_82. Then formula_83, and it is possible to diagonalise formula_3 with a unitary similarity matrix formula_85 that commutes with formula_86, giving formula_87, where formula_88 is a diagonal unitary matrix of phases formula_89. Putting formula_90, one can then re-write the polar decomposition as
so formula_1 then thus also has a spectral decomposition
with complex eigenvalues such that formula_94 and a unitary matrix of complex eigenvectors formula_95.
The polar decomposition of a square invertible real matrix formula_1 is of the form
where formula_98 is a positive-definite matrix and formula_99 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 formula_1, the matrix formula_45 is Hermitian and positive semi-definite, and therefore unitarily equivalent to a positive semi-definite "diagonal" matrix. Let then formula_71 be the unitary such that formula_103, with formula_104 diagonal and positive semi-definite.
Case of formula_1 normal.
If formula_1 is normal, then it is unitarily equivalent to a diagonal matrix: formula_107 for some unitary formula_71 and some diagonal matrix formula_109. We can then write
where formula_111 is a diagonal matrix with containing the "phases" of the elements of formula_109, that is, formula_113 or formula_114 an arbitrary complex number with unit magnitude when formula_115.
The polar decomposition is thus formula_116, with formula_3 and formula_4 diagonal in the eigenbasis of formula_1 and with eigenvalues equal to the phases and absolute values of the eigenvalues of formula_1, respectively.
Case of formula_1 invertible.
From the singular-value decomposition, it can be shown that a formula_1 is invertible if and only if formula_45 (equivalently, formula_124) is. Moreover, this is true if and only if the eigenvalues of formula_45 are all not zero.
In this case, the polar decomposition is directly obtained by writing
and observing that formula_127 is unitary. To see this, we can exploit the spectral decomposition of formula_45 to write formula_129.
In this expression, formula_130 is unitary because formula_71 is. To show that also formula_132 is unitary, we can use the SVD to write formula_133, so that 
where again formula_72 is unitary by construction.
Yet another way to directly show the unitarity of formula_127 is to note that, writing the SVD of formula_1 in terms of rank-1 matrices as formula_138, where formula_139are the singular values of formula_1, we have 
which directly implies the unitarity of formula_127 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 formula_1 reads formula_144, with formula_145 unitary matrices, and formula_104 a diagonal, positive semi-definite matrix. By simply inserting an additional pair of formula_72s or formula_71s, we obtain the two forms of the polar decomposition of formula_1:
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 "l"2(N), then |"A"| = {"AA"}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(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 "AA" ≤ "BB" implies "Ker"("B") ⊂ "Ker"("A").
In particular. If "AA" = "BB", then "C" is a partial isometry, which is unique if "Ker"("B") ⊂ "Ker"("C").
In general, for any bounded operator "A",
where ("AA")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"("A") ⊂ "Ker"("U"). Take "P" to be ("AA")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"("AA") = "Dom"("BB") and "AAh" = "BBh" for all "h" ∈ "Dom"("AA"), 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 "AA" is self-adjoint (with dense domain) and therefore allows one to define ("AA")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", 1"B"("P"), for any Borel set "B" in [0, ∞).
Quaternion polar decomposition.
The polar decomposition of quaternions H depends on the unit 2-dimensional sphere formula_154 of square roots of minus one. Given any "r" on this sphere, and an angle −π < "a" ≤ π, the versor formula_155 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 formula_156
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 formula_157, 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:
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using the Frobenius norm. Including the scale factor, the iteration is now
This iteration can again be combined with rescaling. This particular formula has the benefit that it is also applicable to singular or rectangular matrices "A".