In mathematics, the polar decomposition of a square real or complex matrix
is a positive semi-definite symmetric matrix in the real case), both square and of the same size.
is a complex number with unit norm (an element of the circle group).
The polar decomposition then can be seen as expressing the linear transformation defined by
are unitary matrices (orthogonal if the field is the reals
The polar decomposition of a square invertible real matrix
This makes the derivation of its polar decomposition particularly straightforward, as we can then write
and having eigenvalues equal to the phases and absolute values of those of
[6] In this case, the polar decomposition is directly obtained by writing
is again a diagonal positive semi-definite square matrix with dimensions
As an explicit example of this more general case, consider the SVD of the following matrix:
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.
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.
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(B) ⊂ ker(A).
where (A*A)1/2 is the unique positive square root of A*A given by the usual functional calculus.
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).
A similar but weaker statement holds for the partial isometry: U is in the von Neumann algebra generated by A.
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 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, ∞).
The norm t of a quaternion q is the Euclidean distance from the origin to q.
In the Cartesian plane, alternative planar ring decompositions arise as follows:
Since the operation of multiplying by j reflects a point across the line y = x, the conjugate hyperbola has branches traced by jeaj or −jeaj.
Polar decomposition of an element of the algebra M(2,R) of 2x2 real matrices uses these alternative planar decompositions since any planar subalgebra is isomorphic to dual numbers, split-complex numbers, or ordinary complex numbers.
[8][9] 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: