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Topic: Orthogonal transformations

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	Orthogonal group - Wikipedia, the free encyclopedia
		In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication.
		In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.
		As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.
	en.wikipedia.org /wiki/Orthogonal_group (1412 words)

	Orthogonal matrix - Wikipedia, the free encyclopedia
		However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.
		The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of O(n) of index 2, the special orthogonal group SO(n) of rotations.
		Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a coset; it is also (separately) connected.
	en.wikipedia.org /wiki/Orthogonal_matrix (2765 words)

	GameDev.net - Quaternions and Orthogonal 4x4 Real Matrices
		Orthogonal Matrices ------------------- An _orthogonal_ matrix O is a nonsingular (det(O)/=0) _real_ matrix such that O^-1 = O' -- i.e., its inverse is equal to its transpose.
		Trivial properties of orthogonal matrices: * the transpose and inverse of an orthogonal matrix are themselves orthogonal.
		Now the 4D vector a-b is orthogonal to a+b, becaue a-b and a+b are the diagonals of a rhombus; similar reasoning proves that 1+ab is orthogonal to 1-ab.
	www.gamedev.net /reference/articles/article428.asp (1849 words)

	System and method for orthogonal image transformation - Patent 4736442
		Image data is transformed by transforming image prism size tiles of data using the image prism, and moving them to their final destination in bitmap memory.
		For an orthogonal transformation, the tiles naturally form groups of one, two or four members each, each member of a group having a final destination at a location of another member of the group.
		The set of images and corresponding transformations are identity 106, clockwise rotation by 90 degrees 108, rotation by 180 degrees 110, counterclockwise rotation by 90 degrees 112, mirror along the x-axis 114, mirror along the minor axis 116, mirror along the y-axis 118, and mirror along the major axis 120.
	www.freepatentsonline.com /4736442.html (7361 words)

	Stata help for rotatemat
		If you are interested in rotation after factor, factormat, pca or pcamat, you should consult factor postestimation and pca postestimation, and the general description of rotate as a postestimation facility.
		For orthogonal rotations, quartimax is equivalent to cf(0) and to oblimax.
		When restricted to orthogonal transformations, the oblimin() family is equivalent to the orthomax criterion function.
	www.stata.com /help.cgi?rotatemat (1651 words)

	CDMA subtractive demodulation - Patent 5218619
		In a preferred embodiment of the invention, the composite signal is decoded using iterative orthogonal transformations with a set of codewords to generate a plurality of transformation components associated with the codewords.
		During the iterative process, periodic orthogonal transformations are performed on the remaining portion of the composite signal using at least one of the codewords involved in an earlier transformation.
		After inverse transformation, the samples are rescrambled in a rescrambler 84 using the scrambling code previously used by the descrambler and returned to the buffer 68 through a recirculation loop 86.
	www.freepatentsonline.com /5218619.html (11770 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Algorithms for such problems are sometimes implemented with minor variations in the way elementary hyperbolic transformations are used to construct a $\Sigma$-orthogonal transformation.
		To characterize the possible elementary transformations which can be used to solve a given problem and to clarify important conditioning issues, we introduce a canonical decomposition of a partitioned $\Sigma$-orthogonal matrix which is analogous to the CS decomposition of a partitioned orthogonal matrix.
		We then proceed to prove optimality properties of the hyperbolic transformations given by the decomposition and show how these properties relate to the sensitivity of Cholesky downdating and block Toeplitz factorization problems.
	www-sccm.stanford.edu /~stewart/abstracts/reflect.html (139 words)

	[No title]
		Basically, we transform our repeated measures (levels of the within-subject factor) into (orthogonal) contrasts and either (a) run separate univariate F tests, whose results are then averaged or (b) run a general MANOVA on these contrasts (treating them as a bundle of dependent variables).
		Ideally, your contrasts are orthogonal and refer to the whole pattern of variables (corresponding, for example, to polynomials), in which case they can be directly tested in your SPSS profile run (with appropriate pre-chosen or family-wise alpha settings if you are afraid of Type I errors).
		This also demonstrates the ultimate uselessness of non-orthogonal transformations: they have an effect only on the univariate F's, which are notoriously hard to interpret when they are non-orthogonal (i.e., correlated).
	darkwing.uoregon.edu /~bfmalle/613/L14.html (2571 words)

	Maths - Martin Baker
		In the case of rotations we are interested in the properties of orthogonal matrices.
		In order to represent transforms we derive the sftranslation class which encapsulates the behaviour of 4x4 matrices.
		Ways to represent affine transformations such as angle + vector, multivectors and 4*4 matrices the advantages and uses of each of these representations and how to convert between them.
	www.euclideanspace.com /maths (575 words)

	Orthogonal Transformations (Site not responding. Last check: 2007-10-08)
		If the basis is both orthogonal and normalized then the basis is said to be orthonormal.
		Orthogonal Transformations: For purposes of illustration a 3-dimensional Cartesian coordinate system will be used.
		This expression is known as the orthogonality condition.
	www.geocities.com /physics_world/ma/orthog_trans.htm (515 words)

	dgghrd
		N (input) The order of the matrices A and B. ILO (input) It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to DGGBAL; otherwise they should be set to 1 and N respec- tively.
		If COMPQ='I': on entry, Q need not be set, and on exit it contains the orthogonal matrix Q, where Q' is the product of the Givens transformations which are applied to A and B on the left.
		If COMPZ='I': on entry, Z need not be set, and on exit it contains the orthogonal matrix Z, which is the product of the Givens transformations which are applied to A and B on the right.
	docs.sun.com /source/819-0497/dgghrd.html (515 words)

	Symmetric, Skew-Symmetric, Orthogonal Matrices
		When the orthogonal matrix is a rotation, the interpretation is that the vectors maintain their relationship to each other if they are both rotated.
		Figure 9-2: The Symmetric (complex Hermitic), Skew-Symmetric (complex Skew-Hermitian), Orthogonal, and Unitary Matrix sets characterized by the position of their eigenvalues in the complex plane.
		Multiplication of a vector by an orthogonal matrix is equivalent to an orthogonal geometric transformation on that vector.
	pruffle.mit.edu /3.016/Lecture_09_web/node2.html (285 words)

	Walsh Functions
		Clearly the transform mapping the original sequence into the entries of the bottom row is orthogonal (since it was obtained by a succession of orthogonal transformations).
		Therefore the different patterns which are the columns of this transformation are orthogonal.
		Since all transformations were orthogonal we must have that the collection of vectors corresponding to this choice of patterns is an orthogonal basis of
	www.math.yale.edu /pub/wavelets/software/xwpl/html/manual/node31.html (420 words)

	Documentation for EISPACK
		ELTRAN Accumulates the stabilized elementary similarity transformations used in the reduction of a real general matrix to upper Hessenberg form by ELMHES.
		ORTRAN Accumulates orthogonal similarity transformations in reduction of real general matrix by ORTHES.
		TRED2 Reduce real symmetric matrix to symmetric tridiagonal matrix using and accumulating orthogonal transformation TRED3 Reduce real symmetric matrix stored in packed form to symmetric tridiagonal matrix using orthogonal transformations.
	orion.math.iastate.edu /docs/cmlib/eispack.html (1079 words)

	Similarity Transformations
		The two vectors must transform from the ``old'' to the ``new'' coordinates by:
		Because the transformation matrices are inverses, the following relationship between similar matrices in the old and new coordinate systems is:
		The matrix transformation that takes a coordinate system into its eigenstate is of great interest because it simplifies the mathematical representation of the physical system.
	pruffle.mit.edu /3.016/Lecture_10_web/node1.html (973 words)

	lapack-s/sgghrd.html (Site not responding. Last check: 2007-10-08)
		On exit, the upper triangle and the first subdiago- nal of A are overwritten with the Hessenberg matrix H, and the rest is set to zero.
		If COMPQ='V', then the Givens transformations which are applied to A and B on the left will be applied to the array Q on the right.
		If COMPZ='V', then the Givens transformations which are applied to A and B on the right will be applied to the array Z on the right.
	www.math.utah.edu:8080 /software/lapack/lapack-s/sgghrd.html (394 words)

	Orthogonal Matrices (Site not responding. Last check: 2007-10-08)
		An orthogonal matrix does not change the length of a vector in the L2 norm.
		A similarity transform of a symmetric matrix by an orthogonal matrix will still be symmetric: B=1/QAQ is symmetric if A is and Q is orthogonal.
		The fact that orthogonal matrices don't change the lengths of vectors make them very desirable in numerical applications since they will not increase rounding errors significantly.
	www.cs.colorado.edu /~mcbryan/3656.04/mail/51.htm (181 words)

	Geometry -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		Other geometries can be constructed by choosing a new underlying space to work with (Euclidean geometry uses (A space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional) Euclidean space, R
		As an example, in (The geometery of affine transformations) affine geometry any linear transformation is allowed, and so the first three figures are all congruent; distances and angles are no longer invariants, but colinearity is.
		A (additional info and facts about discrete) discrete form of geometry is treated under (additional info and facts about Pick's theorem) Pick's theorem.
	www.absoluteastronomy.com /encyclopedia/g/ge/geometry.htm (315 words)

	Principal Component Rotation
		Orthogonal transformations can be used on principal components to obtain factors that are more easily interpretable.
		The principal components are uncorrelated with each other, the rotated principal components are also uncorrelated after an orthogonal transformation.
		Different orthogonal transformations can be derived from maximizing the following quantity with respect to
	v8doc.sas.com /sashtml/insight/chap40/sect4.htm (163 words)

	[No title]
		COMPZ (input) CHARACTER1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the orthogonal* matrix Z is returned; = 'V': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned.
		N (input) INTEGER The order of the matrices A and B. ILO (input) INTEGER IHI (input) INTEGER It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to DGGBAL; otherwise they should be set to 1 and N respectively.
		If COMPZ='V': on entry, Z must contain an orthogonal matrix Z1, and on exit this is overwritten by Z1*Z. LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
	www.netlib.org /clapack/double/dgghrd.c (396 words)

	The application of matrix methods to coordinate transformations occurring in systems studies involving large motions of ... (Site not responding. Last check: 2007-10-08)
		The purpose of this paper is to show the method and advantages of matrix algebra in setting up the geometric aspects of problems of airplane motion.
		The various coordinate systems are related to each other by orthogonal transformations in matrix form, and the parameters defining the transformations are found in terms of the dynamical variables of the problem with the help of the transformation matrices.
		The second part shows how to use orthogonal transformations in matrix form by applying them in several examples.
	naca.larc.nasa.gov /reports/1957/naca-tn-3968 (214 words)

	dlatrz(l): factor M-by- real upper trapezoidal ... - Linux man page
		Z is an (M+L)-by-(M+L) orthogonal matrix and, R and A1 are M-by-M upper triangular matrices.
		On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized.
		The kth transformation matrix, Z(k), which is used to introduce zeros into the (m - k + 1)th row of A, is given in the form
	www.die.net /doc/linux/man/manl/dlatrz.l.html (312 words)

	Pauli Quaternions
		Mathematically a rotation is a linear transformation that leaves the norm invariant. These are called orthogonal transformations.
		Conversely any orthogonal transformation can be effected thru multiplication by unimodular quaternions.
		For relativity applications we restrict the orthogonal transformations to those that transform one Hermitian quaternion to another Hermitian quaternion: These are the Lorentz transformations.
	home.pcisys.net /~bestwork.1/quaterni2.html (335 words)

	pdlatrz triangular form by means of orthogonal transformations (Site not responding. Last check: 2007-10-08)
		There is currently no version of the SDSM library with support for a default integer size of 8 bytes (64 bits).
		The upper trapezoidal matrix sub(A) is factored as sub(A) = (R 0) * Z, where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix.
		IA (global input) INTEGER The row index in the global array A indicating the first row of sub(A).
	www.uni-kiel.de /rz/nvv/altix-doc/man_html/man3/pdlatrz.3s.html (732 words)

	eisdoc.f
		Accepts C a pair of real general matrices and reduces C one of them to upper Hessenberg and the other C to upper triangular form using orthogonal C transformations.
		Accepts C a matrix in quasi-triangular form and another C in upper triangular and computes the C eigenvectors of the triangular problem C and transforms them back to the original C coordinates Usually preceded by QZHES, QZIT, C QZVAL.
		C C ELTRAN - - Accumulates the stabilized elementary C similarity transformations used in the C reduction of a real general matrix to upper C Hessenberg form by ELMHES.
	www.cs.yorku.ca /~roumani/fortran/slatecAPI/eisdoc.f.html (1320 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The upper trapezoidal matrix A is fac tored as A = (R 0) * Z, where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix.
		M (input) INTEGER The number of rows of the matrix A. N (input) INTEGER The number of columns of the matrix A. A (input/output) REAL array, dimension (LDA,N) On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized.
		The elements of R are returned in the upper triangular part of A. Z is given by Z = Z(1) * Z(2) *...
	www.ibiblio.org /gferg/ldp/man/manl/stzrzf.l.html (397 words)

	Multi-Channel Remote Sensing Data And Orthogonal Transformations For Change Detection (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: This paper describes the multivariate alteration detection (MAD) transformation which is based on the established canonical correlation analysis.
		It also proposes post-processing of the change detected by the MAD variates by means of maximum autocorrelation factor (MAF) analysis.
		As opposed to most other multivariate change detection schemes the MAD and the combined MAF/MAD transformations are invariant to affine transformations of the originally measured variables.
	citeseer.ist.psu.edu /56095.html (289 words)

	Locking .
		The first instance to discuss is the locking of a single converged Ritz value.
		However, in order to accomplish this, it will be necessary to arrange a transformation of the current Lanczos factorization to one with a small subdiagonal to isolate
		with all the subsequent orthogonal transformations associated with implicit restart applied to
	www.cs.utk.edu /~dongarra/etemplates/node126.html (104 words)

	Linear Algebra
		3/14: orthogonal projections, orthogonal complements and least squares.
		Also, the second lab project was distributed and is due on Friday 3/14.
		Continued discussion of the image of a linear transformation and notion of kernel.
	www.math.utah.edu /~jfernand/teaching/2270/spring03 (776 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		the coordinates of the orthogonal system: identifiers, indexed identifiers, or arithmetical expressions.
		The coordinate systems EllipticCylindrical and Torus are defined with a constant parameter c which has to be passed as an additional argument.
		For computing the components of the gradient with respect to an orthogonal system, it is sufficient to know the ’scale parameters’:
	www.sciface.com /STATIC/DOC30/eng/linalg_ogCoordTab.html (874 words)

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