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Lie group decompositions - Wikipedia, the free encyclopedia The Levi decomposition is a general fact of Lie algebra theory, giving rise to decompositions of semidirect product type for groups, as extensions of a solvable group by a semisimple group. The Iwasawa decomposition KAN of a semisimple group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (a consequence of orthogonalization). The Bruhat decomposition into cells can be regarded as a general expression of the principle of Gauss-Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices - but with exceptional cases. www.wikipedia.org /wiki/Iwasawa_decomposition   (296 words)

Polar decomposition - Wikipedia, the free encyclopedia In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to polar decomposition of a nonzero complex number z The polar decomposition of complex matrix A is a matrix decomposition of the form 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 self-adjoint operator. www.wikipedia.org /wiki/Polar_decomposition   (514 words)

Polyester decomposition process and polyester monomerization system - US Patent 6214893   (Site not responding. Last check: 2007-10-13) This process for the decomposition of a polyester is characterized in that the polyester is decomposed by solvolysis with methanol to form a dimethyl carboxylate and a dihydroxy compound, and these decomposition products are added to and mixed with a melt of the polyester for the purpose of decomposing the polyester by solvolysis. A process for the decomposition of a polyester as claimed in claim 1 wherein the polyester is decomposed by solvolysis with a lower alkyl alcohol and the dialkyl carboxylate and dihydroxy compound obtained as decomposition products by solvolysis with the lower alkyl alcohol are recycled as solvents for the polyester. A process for the decomposition of a polyester as claimed in claim 1, wherein the polyester is selected from the group consisting of polyethylene terephthalate, polybutylene terephthalate, polycyclohexanedimethylene terephthalate, polyethylene naphthalate, polybutylene naphthalate, and polycarbonate. www.patentstorm.us /patents/6214893.html   (2888 words)

PlanetMath: derivation of rotation matrix using polar coordinates "derivation of rotation matrix using polar coordinates" is owned by stevecheng. See Also: rotation matrix, polar coordinates, decomposition of orthogonal operators as rotations and reflections, derivation of 2D reflection matrix This is version 3 of derivation of rotation matrix using polar coordinates, born on 2005-07-25, modified 2005-07-25. planetmath.org /encyclopedia/DerivationOfRotationMatrixUsingPolarCoordinates.html   (146 words)

Polar/4 - All About All findings   (Site not responding. Last check: 2007-10-13) The Polar Medal is a medal awarded by the Government of the United Kingdom, originally instituted in 1857 as the Arctic Medal to reward earlier explorers attempting to discover the North-West Passage. The cold temperatures in the polar regions cause air to descend to create the high pressure (a process called subsidence), just as the warm temperatures around the equator case air to rise to create the low pressure doldrums.Surface temperatures under the polar highs are the coldest on Earth, with no... In probability theory, the polar distribution is the probability distribution of angles occurring in a set of two-dimensional vectors, denoted by It is usually graphically represented as a closed curve, where the radius. www.allaboutall.info /search/Polar/4   (712 words)

Process for the production of a phenol and a carbonyl compound by the catalyzed decomposition of an aromatic ... A process according to claim 1 wherein a liquid cleavage decomposition catalyst is fed continuously to said reactor and is removed continuously from said reactor in said liquid phase with said higher-boiling residues. Whilst organic acids such as carboxylic acids may be used as the liquid phase cleavage decomposition catalyst it is preferred to employ a mineral acid catalyst, such as sulphuric acid or phosphoric acid, of which sulphuric acid is preferred. Preferably the catalyst is sulphuric acid, which is suitably fed to the cleavage decomposition reaction as a dilute solution in, for example, one or more of acetone or acetophenone or cumene in an amount of from 50 to 2000 ppm of free acid, based on the weight of the feed. www.freepatentsonline.com /4246203.html   (4183 words)

SISC Volume 7 Issue 4 A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. To add to the known best approximation property of the unitary polar factor, the Hermitian polar factor $H$ of a nonsingular Hermitian matrix $A$ is shown to be a good positive definite approximation to $A$and $\frac{1}{2}(A + H)$ is shown to be a best Hermitian positive semi-definite approximation to $A$. Applications of the polar decomposition to factor analysis, aerospace computations and optimisation are outlined; and a new method is derived for computing the square root of a symmetric positive definite matrix. locus.siam.org /SISC/volume-07/art_0907079.html   (189 words)

Dynamical Methods for Polar Decomposition and Inversion of Matrices   (Site not responding. Last check: 2007-10-13) We show how one may obtain polar decomposition as well as inversion of fixed and time-varying matrices using a class of nonlinear continuous-time dynamical systems. We then construct a dynamical system that solves for the polar decomposition factors of a time-varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition. Finially, using homotopy again, we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time. www.cds.caltech.edu /~marsden/bib/1997/07-GeMa1997   (197 words)

A Refined Polar Decomposition: A=UPD A refinement of the polar decomposition of a nonsingular matrix A is considered. It is shown that such a decomposition exists and is unique. The refined polar decomposition can be used to parameterize the orbit of a matrix with distinct eigenvalues. epubs.siam.org /sam-bin/dbq/article/36921   (133 words)

Entropie/(/Anisotropie) Decomposition The Cloud decomposition theorem is widely used and a lot of publication about the application of this decomposition can be found in literature. As said before, this decomposition theorem is based on the coherency matrix. This is a crucial step for this decomposition approach, since - depending on your implementation, programming language, and computer system, the calculation of the eigenvalues and vectors for a scene can take any time between some minutes and some hours. epsilon.nought.de /tutorials/polsmart/node38.html   (1201 words)

Citations: Polar decomposition and matrix sign function condition estimates - Kenney, Laub (ResearchIndex)   (Site not responding. Last check: 2007-10-13) Polar decompositions and matrix sign function condition estimates. Polar decomposition and matrix sign function condition estimates. Mathias [30] derived an expression for the Fr echet derivative using the Schur decomposition. citeseer.lcs.mit.edu /context/152240/0   (1242 words)

Modal Interpretations of Quantum Mechanics   (Site not responding. Last check: 2007-10-13) We see in the biorthogonal decomposition theorem a way to choose the possible value states from one single decomposition of the (mixed) dynamical state of a system: let them be (for each component system) the elements of the basis picked out by the theorem. For those interpretations that say little beyond the application of these decompositions to determine the set of possibly-possessed properties, the question is all the more pressing, and the interpretations are all the more lacking in direct physical motivation. In that case, one might expect that the spectral decompositions (of the reduced states for the apparatus and measured system) would pick out states for the two systems that are extremely close to the ‘ideal’ states. plato.stanford.edu /entries/qm-modal   (7470 words)

Polarization References Coulson, Kinsell L. Polarization and Intensity of Light in the Atmosphere, A. DEEPAK Publishing, 1988, 596 pp., ISBN 0-937194-12-3. Polarization and Color Techniques in Industrial Inspection; Elzbieta A. Marszalec, Emanuele Trucco; Eds. Proceedings of SPIE v 3754, Proceedings of the 1999 Polarization: Measurement, Analysis, and Remote Sensing II, p 108-117. www.cis.rit.edu /~jrs6394/Research/bib/polar_bib.htm   (2918 words)

[No title]   (Site not responding. Last check: 2007-10-13) Pawel Zielinski and Krystyna Zietak "The polar decomposition - properties, applications and algorithms" In the paper we review the numerical methods for computing the polar decomposition of a matrix. Moreover, the applications of the polar decomposition and the most important its properties are mentioned. Keywords and phrases: polar decomposition, singular value decomposition, complete orthogonal decomposition, approximation of matrices, Procrustes problem, matrix square root, matrix sign, optimal scaling, acceleration, Schultz iteration, Newton's method www.im.pwr.wroc.pl /~zietak/papers/polar.txt   (72 words)

College Mathematics Journal, The: A fresh approach to the singular value decomposition   (Site not responding. Last check: 2007-10-13) Even Dan Kalman, in his delightful article [7], concluded that "it is probably not feasible to include the SVD in the first linear algebra course." We aim to convince the reader that the singular value decomposition is in fact a very natural and approachable topic. It is equivalent, in the case of invertible square matrices, to the easy to motivate (and establish) polar decomposition. It is easy to write a short program to automate finding such polar decompositions using one of the standard software packages; we offer a short Matlab routine later on. www.looksmartcollege.com /p/articles/mi_qa3773/is_199805/ai_n8802131   (1338 words)

Polar Decomposition of a Matrix and Method of Square Roots Polar Decomposition of a Matrix and Method of Square Roots in the form (12) is called the polar decomposition. Hence the polar decomposition of the matrix A is www.cs.ut.ee /~toomas_l/linalg/lin2/node26.html   (254 words)

e-Prints Soton - A Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry A Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry Poladian, L., Issa, N.A. and Monro, T.M. A Fourier decomposition algorithm for leaky modes of fibres with arbitrary geometry. A new algorithm for calculating the confinement loss of leaky modes in arbitrary fibre structures is presented within the scalar wave approximation. eprints.soton.ac.uk /13775   (141 words)

Polar Decomposition Theory in Nonlinear Analyses of Solids and Structures   (Site not responding. Last check: 2007-10-13) Polar Decomposition Theory in Nonlinear Analyses of Solids and Structures The physical meaning, geometric interpretation, and applications of the polar decomposition theory and Jaumann strains and stresses in nonlinear modeling and analysis of solids and structures are considered. Jaumann strains and stresses prove to be objective geometric measures defined with respect to the deformed system configuration, and constant material stiffnesses obtained from experiments using engineering stresses and strains can be directly used in the constitutive equation. www.pubs.asce.org /WWWdisplay.cgi?9501158   (161 words)

[No title]   (Site not responding. Last check: 2007-10-13) wrote: :Hello all, : :I would like to have a reference to the real polar decomposition which :states that every real n*n matrix A has a representation A=H*U where H :is symmetric and positive semidefinite and U is orthogonal. Here is a sledgehammer approach (cracking a walnut with a steamroller): The singular value decomposition of a real matrix A is A = U * S * V' (V' means the transpose of V) where U, V are real orthogonal and S is diagonal, with entries non-negative and sorted in descending order along the diagonal. If you look at the proof then you can see that in the real case AA* is real and A*A real, hence their eigenvectors can be chosen real and you are done. www.math.niu.edu /~rusin/known-math/99/real_polar   (282 words)

ACC00-IEEE1278   (Site not responding. Last check: 2007-10-13) Robust stability of the convex hull of matrices using the polar decomposition The result are based on the polar decomposition and a novel sufficient condition for the stability of a given matrix, i.e. a matrix is stable if the unitarian matrix in its polar decomposition is stable. nt1.rsip.lsu.edu /cebopenweb/conferences/acc2K/T-Program/program/Manuscripts/ACC00-IEEE1278.htm   (65 words)

Perturbation Bounds For The Polar Decomposition - Mathias (ResearchIndex)   (Site not responding. Last check: 2007-10-13) ...in this paper are motivated by the problem of computing the norm of the derivative of the polar factor in the polar decomposition. 10 Polar decomposition and matrix sign function condition estim.. 7 Perturbation bounds on the polar decomposition (context) - Barrlund - 1989 citeseer.ifi.unizh.ch /mathias97perturbation.html   (525 words)

Computing The Polar Decomposition And The Matrix (ResearchIndex)   (Site not responding. Last check: 2007-10-13) Sign Decomposition In Matrix Groups Nicholas J. Higham, D. Steven Mackey,... Abstract: For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of A # G also lie in G; moreover, the polar factors of A lie in G if the matrix of the underlying form is unitary. 3 Polar decompositions in finite dimensional indefinite scalar.. citeseer.lcs.mit.edu /625298.html   (472 words)

Nick Higham's Papers on Matrix Functions and Nonlinear Matrix Equations   (Site not responding. Last check: 2007-10-13) Computing the Polar Decomposition and the Matrix Sign Decomposition in Matrix Groups (With D. Steven Mackey, Niloufer Mackey and Françoise Tisseur), SIAM J. Matrix Anal. The Matrix Sign Decomposition and its Relation to the Polar Decomposition, Linear Algebra and Appl., 212/213:3-20, 1994. Fast polar decomposition of an arbitrary matrix (with R. Schreiber), SIAM J. Sci. www.ma.man.ac.uk /~higham/pap-mf.html   (287 words)

Abstracts   (Site not responding. Last check: 2007-10-13) It turns out that symmetric spaces have many interesting applications in Geometric Integration, such as the classification of sandwich products (a la McLachlan/Quispel), generalizing polar decompositions, the study of time reversal symmetry and providing a framework for studying numerical integrators on symmetric spaces (e.g. Generalization of the polar decomposition to symmetric spaces It is well known in linear algebra that an arbitrary real matrix A can be written as A=HU, where H is a symmetric positive (semi-)definite matrix and U is orthogonal. www.ii.uib.no /~anto/abstracts.html   (411 words)

► » Polar Decomposition I need polar decomosition for dynamic graph drawing. decomposition method for 2x2 matrices described in "Matrix Animation and Polar Decomposition" by K. Shoemake and T. Duff: www.science-chat.org /Polar-Decomposition-5217391.html   (100 words)

Topics: P Polar Decomposition Theorem > see examples of lie groups [SL(2,C)]. Polarization in Electricity and Field Theory > see electricity; qft states; vacuum. Idea: A polarization is an n-dimensional completely degenerate subspace of a symplectic vector space, or integrable distribution on a 2n-dimensional symplectic manifold (it thus forms Lagrangian submanifolds). www.phy.olemiss.edu /~luca/Topics/p.html   (2437 words)

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