A polar decomposition of 
with 
is a factorization 
, the place 
has orthonormal columns and 
is Hermitian constructive semidefinite. This decomposition is a generalization of the polar illustration 
of a fancy quantity, the place 
corresponds to 
and 
to 
. When 
is actual, is symmetric constructive semidefinite. When 
, is a sq. unitary matrix (orthogonal for actual ).
We have now 
, so 
, which is the distinctive constructive semidefinite sq. root of 
. When has full rank, is nonsingular and 
is exclusive, and on this case could be expressed as
An instance of a polar decomposition is
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For an instance with a rank-deficient matrix contemplate
for which 
and so 
. The equation then implies that
so isn’t distinctive.
The polar issue has the necessary property that it’s a closest matrix with orthonormal columns to in any unitarily invariant norm. Therefore the polar decomposition gives an optimum solution to orthogonalize a matrix. This technique of orthogonalization is utilized in varied purposes, together with in quantum chemistry, the place it’s known as Löwdin orthogonalization. Most frequently, although, orthogonalization is finished by way of QR factorization, buying and selling optimality for a sooner computation.
An necessary utility of the polar decomposition is to the orthogonal Procrustes downside
the place 
and the norm is the Frobenius norm 
. This downside, which arises in issue evaluation and in multidimensional scaling, asks how carefully a unitary transformation of 
can reproduce . Any answer is a unitary polar issue of 
, and there’s a distinctive answer if is nonsingular. One other utility of the polar decomposition is in 3D graphics transformations. Right here, the matrices are 
and the polar decomposition could be computed by exploiting a relationship with quaternions.
For a sq. nonsingular matrix , the unitary polar issue could be computed by a Newton iteration:
The iterates 
converge quadratically to . This is only one of many iterations for computing and far work has been executed on the environment friendly implementation of those iterations.
If 
is a singular worth decomposition (SVD), the place 
has orthonormal columns, 
is unitary, and 
is sq. and diagonal with nonnegative diagonal components, then
the place has orthonormal columns and is Hermitian constructive semidefinite. So a polar decomposition could be constructed from an SVD. The converse is true: if is a polar decomposition and 
is a spectral decomposition (
unitary, 
diagonal) then 
is an SVD. This latter relation is the premise of a way for computing the SVD that first computes the polar decomposition by a matrix iteration then computes the eigensystem of , and which is extraordinarily quick on distributed-memory manycore computer systems.
The nonuniqueness of the polar decomposition for rank poor , and the dearth of a passable definition of a polar decomposition for 
, are overcome within the canonical polar decomposition, outlined for any 
and 
. Right here, with a partial isometry, is Hermitian constructive semidefinite, and 
. The superscript “
” denotes the Moore–Penrose pseudoinverse and a partial isometry could be characterised as a matrix for which 
.
Generalizations of the (canonical) polar decomposition have been investigated during which the properties of and are outlined with respect to a normal, probably indefinite, scalar product.
References
It is a minimal set of references, which comprise additional helpful references inside.
- Nicholas J. Higham, Features of Matrices: Idea and Computation, Society for Industrial and Utilized Arithmetic, Philadelphia, PA, USA, 2008. (Chapter 8.)
- Nicholas Higham, Christian Mehl, and Françoise Tisseur, The canonical generalized polar decomposition, SIAM J. Matrix Anal. Appl. 31(4), 2163–2180, 2010.
- Nicholas J. Higham and Vanni Noferini, An algorithm to compute the polar decomposition of a
matrix, Numer. Algorithms 73, 349–369, 2016.
- Yuji Nakatsukasa and Nicholas J. Higham, Steady and environment friendly spectral divide and conquer algorithms for the symmetric eigenvalue decomposition and the SVD, SIAM J. Sci. Comput. 35(3), A1325–A1349, 2013.
- Dalal Sukkari, Hatem Ltaief, Aniello Esposito, and David Keyes, A QDWH-based SVD software program framework on distributed-memory manycore programs, ACM Trans. Math. Software program 45, 18:1–18:21, 2019.
