Polar decomposition
Polar decomposition is a concept from linear algebra and functional analysis, both branches of mathematics. It refers to a particular decomposition into a product of matrices with real or complex entries, and generalization of linear operators on a Hilbert space. The polar decomposition of matrices and operators generalizes the polar decomposition of a non-zero complex number z in the product of its magnitude and number on the complex unit circle, with the argument of z, ie:
Polar decomposition real or complex matrices
If A is a square matrix, so is called (left ) polar decomposition is a factorization, where
- U is an orthogonal and P is a positive semidefinite symmetric matrix is the real case, ie, and;
- In the complex case U is a unitary and P is a positive semidefinite Hermitian matrix, ie, and.
If A is invertible, then the decomposition is unique, and U and -U are the orthogonal or unitary matrices with the smallest or largest distance to A.
Calculating the polar decomposition
( The real methods are a special case of the complex, the adjoint matrices are then equal to the transposed matrices. )
On the Singular Value Decomposition
With the singular value decomposition can be the polar decomposition as and determine.
As an iterative determination of the symmetric factor
P can be used as the uniquely determined positive semidefinite square root of being diagnosed. For this, the Heron's root method can be generalized and. If A is invertible, then the method converges with limit P and.
As an iterative determination of the orthogonal factor
Another derived from the Hero 's square root method determines the unitary factor U as the limit of recursion
This is locally quadratically convergent. To accelerate the global convergence, especially if all the singular values of A or all are very large very small, rescaled to the iteration
Wherein should be close to the geometric center of the singular values of the and and its inverse can be estimated by a combination of different matrix norms. Have been proposed, inter alia, the factors
With the row and column sum norms and
With the Frobenius norm.
Polar decomposition of operators
Abstract one defines:
A ( left and right ) polar decomposition of a continuous linear operator A to a Hilbert space, that is, is one of the following multiplicative decompositions:
Here are positive operators, which are formed by the continuous functional calculus, and is a partial isometry, that is.
By a theorem of functional analysis is available for every continuous linear operator A on a Hilbert space H such polar decomposition.
Instead you write well. If is invertible, then also and is unitary.
Example of use
In continuum mechanics, the " polar decomposition " of the deformation gradient finds application in the description of deformation and strain tensors defined it.
Related Topics
- Singular value decomposition