Polar decomposition

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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.[1]

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 .[2] This decomposition is useful in computing the fundamental group of (matrix) Lie groups.[3]

The polar decomposition can also be defined as where is symmetric positive-definite but is in general a different matrix, while 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 as , where is its absolute value (a non-negative real number), and is a complex number with unit norm (an element of the circle group).

Properties[edit]

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.[4] If A is invertible, then P is positive-definite, thus also invertible and the matrix U is uniquely determined by

Intuitive interpretation[edit]

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 (the action of ), followed by a single rotation or reflection of (the action 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[edit]

In terms of the singular value decomposition (SVD) of , , one has

where , , and are unitary matrices (called orthogonal matrices if the field is the reals ). 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[edit]

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 normal[edit]

If is normal, then it is unitarily equivalent to a diagonal matrix: for some unitary and some diagonal matrix . We can then write

where 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 invertible[edit]

From the singular-value decomposition, it can be shown that a is invertible if and only if (equivalently, ) is. Moreover, this is true if and only if the eigenvalues of are all not zero[5].

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[edit]

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[edit]

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*AB*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*AB*B implies Ker(B) ⊂ Ker(A).

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[edit]

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 hDom(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[edit]

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[edit]

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

  • If x ≠ 0, z = x(1 + ε(y/x)) is a polar decomposition of a dual number z = x + yε, where ε2 = 0; i.e., ε is nilpotent. In this polar decomposition, the unit circle has been replaced by the line x = 1, the polar angle by the slope y/x, and the radius x is negative in the left half-plane.
  • If x2y2, then the unit hyperbola x2y2 = 1 and its conjugate x2y2 = −1 can be used to form a polar decomposition based on the branch of the unit hyperbola through (1, 0). This branch is parametrized by the hyperbolic angle a and is written

    where j2 = +1 and the arithmetic[6] of split-complex numbers is used. The branch through (−1, 0) is traced by −eaj. Since the operation of multiplying by j reflects a point across the line y = x, the second hyperbola has branches traced by jeaj or −jeaj. Therefore a point in one of the quadrants has a polar decomposition in one of the forms:

    The set { 1, −1, j, −j } has products that make it isomorphic to the Klein four-group. Evidently polar decomposition in this case involves an element from that group.

Numerical determination of the matrix polar decomposition[edit]

To compute an approximation of the polar decomposition A = UP, usually the unitary factor U is approximated.[7][8] 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:

  • Every step or in regular intervals, the range of the singular values of is estimated and then the matrix is rescaled to to center the singular values around 1. The scaling factor is computed using matrix norms of the matrix and its inverse. Examples of such scale estimates are:

    using the row-sum and column-sum matrix norms or

    using the Frobenius norm. Including the scale factor, the iteration is now

  • The QR decomposition can be used in a preparation step to reduce a singular matrix A to a smaller regular matrix, and inside every step to speed up the computation of the inverse.
  • Heron's method for computing roots of can be replaced by higher order methods, for instance based on Halley's method of third order, resulting in
    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.

See also[edit]

References[edit]

  1. ^ Hall 2015 Section 2.5
  2. ^ Hall 2015 Theorem 2.17
  3. ^ Hall 2015 Section 13.3
  4. ^ Hall 2015 Lemma 2.18
  5. ^ Note how this implies, by the positivity of , that the eigenvalues are all real and strictly positive.
  6. ^ Sobczyk, G.(1995) "Hyperbolic Number Plane", College Mathematics Journal 26:268–80
  7. ^ Higham, Nicholas J. (1986). "Computing the polar decomposition with applications". SIAM J. Sci. Stat. Comput. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics. 7 (4): 1160–1174. CiteSeerX 10.1.1.137.7354. doi:10.1137/0907079. ISSN 0196-5204.
  8. ^ Byers, Ralph; Hongguo Xu (2008). "A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability". SIAM J. Matrix Anal. Appl. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics. 30 (2): 822–843. CiteSeerX 10.1.1.378.6737. doi:10.1137/070699895. ISSN 0895-4798.