Polar decomposition

Polar decomposition is a term from linear algebra and functional analysis , both sub-areas of mathematics . It refers to a special decomposition into a product of matrices with real or complex entries, and a generalization of linear operators on a Hilbert space . The polar decomposition of matrices and operators generalizes the polar decomposition of a non-vanishing complex number into the product of its absolute value and a number on the complex unit circle , with the argument of , thus . ${\displaystyle z}$ ${\displaystyle r=|z|}$ ${\displaystyle e^{i\varphi }}$ ${\displaystyle \varphi }$ ${\displaystyle z}$ ${\displaystyle z=r\cdot e^{i\varphi }}$

Polar decomposition of real or complex matrices

If a square matrix is used, a (right) polar decomposition is a factorization ${\displaystyle A}$

${\displaystyle A=U\,P}$,

in which

• in the real case is an orthogonal and a positive semidefinite symmetric matrix and ${\displaystyle U}$ ${\displaystyle P}$
• in the complex case is a unitary and a positive semidefinite Hermitian matrix . ${\displaystyle U}$ ${\displaystyle P}$

If invertible , the decomposition is unambiguous, positively definite and / or are the orthogonal or unitary matrices with the smallest or largest distance to . ${\displaystyle A}$ ${\displaystyle P}$ ${\displaystyle U}$ ${\displaystyle -U}$ ${\displaystyle A}$

Calculation of the polar decomposition

The real methods are a special case of the complex, whereby the adjoint matrix is then equal to the transposed matrix . ${\displaystyle X^{\ast }}$ ${\displaystyle X^{T}}$

About the singular value decomposition

With the singular value decomposition

${\displaystyle A=V\,\Sigma \,W^{*}}$

one can use the polar decomposition as

${\displaystyle U=V\,W^{*}}$ and ${\displaystyle P=W\,\Sigma \,W^{*}}$

determine.

As an iterative determination of the symmetrical factor

The matrix can be viewed as the uniquely determined positive semidefinite square root of ${\displaystyle P}$

${\displaystyle B=A^{\ast }\,A=PU^{*}UP=P^{2}}$

to be determined. To this end, the Heron's root method can be generalized to

${\displaystyle P_{0}=I}$and . ${\displaystyle P_{k+1}={\tfrac {1}{2}}(P_{k}+P_{k}^{-1}B)}$

If invertible, the method converges with limit value and . ${\displaystyle A}$ ${\displaystyle P}$ ${\displaystyle U=A\,P^{-1}}$

As an iterative determination of the orthogonal factor

Another method derived from Heron's root extraction determines the unitary factor as the limit value of the recursion ${\displaystyle U}$

${\displaystyle U_{0}=A}$and . ${\displaystyle U_{k+1}={\tfrac {1}{2}}(U_{k}+(U_{k}^{*})^{-1})}$

This is locally quadratically convergent. To accelerate the global convergence, in particular if all singular values ​​are very large or all very small, the iteration is rescaled ${\displaystyle A}$

${\displaystyle U_{k+1}={\tfrac {1}{2}}(\gamma _{k}U_{k}+(\gamma _{k}U_{k}^{*})^{-1})}$,

where should lie near the geometric center of the singular values ​​of and can be estimated by combinations of different matrix norms of and their inverses . The factors suggested were among others ${\displaystyle \gamma _{k}^{-1}}$ ${\displaystyle U_{k}}$ ${\displaystyle U_{k}}$

${\displaystyle \gamma _{k}={\sqrt[{4\;}]{\frac {\|U_{k}^{-1}\|_{1}\,\|U_{k}^{-1}\|_{\infty }}{\|U_{k}\|_{1}\,\|U_{k}\|_{\infty }}}}}$

with the row and column sum norms and

${\displaystyle \gamma _{k}={\sqrt {\frac {\|U_{k}^{-1}\|_{F}}{\|U_{k}\|_{F}}}}}$

with the Frobenius norm .

Polar decomposition of operators

A (left or right) polar decomposition of a continuous linear operator on a Hilbert space , that is , is one of the following multiplicative decomposition: ${\displaystyle A}$ ${\displaystyle A\in {\mathcal {L}}(H)}$

${\displaystyle A=U{\sqrt {A^{*}A}}={\sqrt {AA^{*}}}\,U}$.

Here, and are positive operators that are formed using the continuous functional calculus , and is a partial isometry , that is . Such a polar decomposition exists for every continuous linear operator on a Hilbert space. Instead of writing, too . If is invertible, so is and is unitary. ${\displaystyle {\sqrt {A^{*}A}}}$ ${\displaystyle {\sqrt {AA^{*}}}}$ ${\displaystyle U\in {\mathcal {L}}(H)}$ ${\displaystyle U^{*}UU^{*}U=U^{*}U}$ ${\displaystyle {\sqrt {A^{*}A}}}$ ${\displaystyle \left|A\right|}$ ${\displaystyle A}$ ${\displaystyle |A|}$ ${\displaystyle U}$

Application example

In continuum mechanics the "polar decomposition" of the place deformation gradient an application in the description of deformations and defined it tensors .

literature

• W. Rudin: Functional Analysis , 2nd Edition, McGraw-Hill, 1991, pp. 330-333.

Individual evidence

1. ↑ Nicholas J. Higham: Computing the polar decomposition with applications Archived from the original on May 8, 2013. Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. In: Society for Industrial and Applied Mathematics (Ed.): SIAM J. Sci. Stat. Comput. . 7, No. 4, 1986, pp. 1160-1174. ISSN 0196-5204 . doi : 10.1137 / 0907079 . Retrieved August 26, 2010.   
2. ^ Ralph Byers, Hongguo Xu: A New Scaling for Newton's Iteration for the Polar Decomposition and its Backward Stability . In: Society for Industrial and Applied Mathematics (Ed.): SIAM J. Matrix Anal. Appl. . 30, No. 2, 2008, pp. 822-843. ISSN 0895-4798 . doi : 10.1137 / 070699895 .