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Topic: Orthonormal matrix

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	Inverting an Orthogonal Matrix (Site not responding. Last check: 2007-10-21)
		Therefore the determinant of an orthonormal matrix is ±1.
		This is an orthonormal matrix with determinant 1.
		Scale the rows of m, and the inverse of an orthogonal matrix is its tranjugate, with its columns scaled, as described above.
	www.mathreference.com /la-det,iorth.html (260 words)

	Station Information - Orthonormal matrix
		In linear algebra, an orthonormal matrix is a (not necessarily square) matrix with real or complex entries whose columns, treated as vectorss in R
		The real n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from an orthogonal matrix; the complex n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from an unitary matrix.
		In particular, unitary and orthogonal matrices are themselves orthonormal.
	www.stationinformation.com /encyclopedia/o/or/orthonormal_matrix.html (123 words)

	Matrix Decompositions
		The LU decomposition of a matrix is frequently used as part of a Gaussian elimination process for solving a matrix equation.
		The first element is a combination of upper and lower triangular matrices, the second element is a vector specifying rows used for pivoting (a permutation vector which is equivalent to the permutation matrix), and the third element is an estimate of the condition number.
		This matrix cannot be diagonalized because the matrix of eigenvectors is singular.
	documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/4.5.html (2005 words)

	Rigid Rotations
		A matrix is orthonormal iff it defines a rigid rotation.
		We already showed the inverse of such a matrix is its transpose, and the inverse of a rigid rotation preserves lengths and angles, hence the inverse, or transpose, of an orthonormal matrix is orthonormal.
		The rest of m is orthonormal, and spans the subspace perpendicular to w.
	www.mathreference.com /la-det,rot.html (1171 words)

	Egwald Mathematics - Linear Algebra: Matrices and Matrix Decomposition
		A matrix of dimension m by 1 is called a column vector; a matrix of dimension 1 by n is called a row vector.
		of a square matrix A of dimension n is a measure of the "distance" of A from the zero matrix, O, with properties similar to those of a vector norm.
		The columns of the orthogonal matrix are divided by their norms to produce an orthonormal matrix.
	www.egwald.com /linearalgebra/matrices.php (3976 words)

	Orthonormal matrix
		In linear algebra, an orthonormal matrix is a (not necessarily square) matrix with real or complex entries whose columns, treated as vectors in R
		The real n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from an orthogonal matrix; the complex n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from a unitary matrix.
		In particular, unitary and orthogonal matrices are themselves orthonormal.
	www.wapipedia.org /wikipedia/mobiletopic.aspx?cur_title=Orthonormal_matrix (158 words)

	PlanetMath: adjoint endomorphism
		Specializing further, suppose that the basis in question is orthonormal, i.e.
		However, the analogous operation at the matrix level is the conjugate transpose.
		Cross-references: orthonormal basis, conjugate transpose, level, operation, complex, unitary space, transpose, orthonormal, matrix, basis, commutative diagram, isomorphism, linear isomorphism, Endomorphism, inner product, real, bilinear mapping, non-degenerate, symmetric, field, vector space, finite-dimensional
	planetmath.org /encyclopedia/Adjoint3.html (175 words)

	LTI-Lib: ltiSVD.h Source File
		U is a 00048 * column-orthonormal matrix, W is a diagonal matrix with the 00049 * singular values on the diagonal, and V is a orthonormal 00050 * matrix.
		U is mn, W is a diagonal 00337 matrix with n elements (which is implemented as vector), V is a 00338 * n*n-matrix.
		If 00341 * src is a mn matrix, U will also be of size mn 00342 * @param w vector with the singular values, sorted descendingly 00343 * The elements of this vector constitute the diagonal 00344 * of the W matrix.
	ltilib.sourceforge.net /doc/html/ltiSVD_8h-source.html (921 words)

	Thermo Scientific - Algorithms - Principal Component Regression
		On the other hand, the scores in the S matrix are unique to each calibration spectrum, and just as a spectrum is represented by a collection of absorbances at a series of wavelengths, it can also be series of scores for a given set of factors.
		Multiplying any matrix by the identity matrix is the same as multiplying a single number by one; the result is always the number again.
		where C is the m by n matrix of constituent concentrations, B is an m byf matrix of the regression coefficients, A is an n by p matrix of spectral absorbances, and F is an f by p matrix of eigenvectors.
	www.thermo.com /com/cda/resources/resources_detail/1,2166,13414,00.html (1150 words)

	Computer Graphics : 3D Transformation : 14 / 19 : Properties of the Rotation Matrix
		is the transpose of A and I is the identity matrix, and
		The first condition means that A is an orthogonal matrix, and the second restricts the determinant to +1 (while a general orthogonal matrix may have determinant -1 or +1).
		In fact, given any orthonormal basis, the matrix whose rows are that basis is an orthogonal matrix.
	escience.anu.edu.au /lecture/cg/Transformation/rotationMatrixProp.en.html (369 words)

	InfoVis CyberInfrastructure- Latent Semantic Analysis
		W is a orthonormal matrix and its rows correspond to the rows of X, but it has m columns corresponding to new, specially derived variables such that there is no correlation between any two columns; i.e., each is linearly independent of the others.
		P is an orthonormal matrix and has columns corresponding to the original columns but m rows composed of derived singular vectors.
		Matrix multiply W (its rows correspond to the terms (rows) of X) by the reduced matrix S. Normalize resulting vectors, i.e., divide each vector by its length.
	iv.slis.indiana.edu /sw/lsa.html (1985 words)

	Subspaces
		the columns of the hanger matrix corresponding to non-zero singular values are an orthonormal basis for Col[A].
		the rows of the aligner matrix corresponding to non-zero singular values are an orthonormal basis for Row[A].
		Since the columns of the hanger matrix corresponding to non-zero singular values form a basis for the column space, you know that the rank of A is equal to the number of non-zero singular values.
	www.uwlax.edu /faculty/will/svd/subspaces (529 words)

	Unit Vector Math for 3D Graphics
		The rotation matrix is derived from both the object's and the observer's unit vectors; the translation vector is derived from the object's position, the observer's position, and the observer's unit vectors.
		As discussed previously, the unit vectors form an Orthonormal matrix; its inverse is simply the Transpose.
		Matrix notation is, after all, only a shorthand for representing simultaneous equations.
	www.jmargolin.com /uvmath/uvmath.htm (3812 words)

	6 Decompositions (Site not responding. Last check: 2007-10-21)
		Matrix decompositions are useful in numerical problems, in particular for solving systems of linear equations.
		is the matrix obtained by the concatenation of the eigenvectors.
		It takes the matrix as its argument, returns in the form of a list the upper and lower triangular matrices, and an index vector which records the row permutations in the LU decomposition:
	www.xplore.jp /tutorials/matrixnode7.html (522 words)

	Hadamard Matrix - Surch (Site not responding. Last check: 2007-10-21)
		It has been a common misapprehension over the last seven or eight years that the number of pairwise non-isomorphic 2-(23,11,5) designs is 1102.
		Hadamard Matrix -- from MathWorld Hadamard Matrix -- from MathWorld A Hadamard matrix is a type of square (-1,1)-matrix invented by Sylvester (1867) under the name of anallagmatic pavement, 26 years before Hadamard (1893) considered them.
		role because of the connection between rows of the Walsh-Hadamard matrix and the codewords of RM(1, m)...
	www.surch.co.uk /hadamard+matrix (372 words)

	PlanetMath: second fundamental form
		Or in matrix form (with respect to the coordinates
		is expressed as a matrix with orthonormal coordinates (with respect to
		Matrix representations of second fundamental form and Weingarten map
	planetmath.org /encyclopedia/WeingartenMap.html (474 words)

	Matrix Formulation of the DFT
		The DFT can be formulated as a complex matrix multiply, as we show in this section.
		Computation of the DFT matrix in Matlab is illustrated in §I.4.3.
		Such a complex matrix is said to be unitary.
	ccrma-www.stanford.edu /~jos/mdft/Matrix_Formulation_DFT.html (207 words)

	Summary for Exam 2
		Thus, every n by m matrix with independent columns is the product of an orthonormal matrix and an upper triangular matrix, and the upper triangular matrix has positive entries on the diagonal.
		The transpose A^T of the m by n matrix A = [a_{ij}] is the n by m matrix [a_{ji}].
		A matrix is called orthogonal if it is a square matrix, say n by n, and its columns are an orthonormal set.
	www.math.lsa.umich.edu /~hochster/419/ex2syl.html (1082 words)

	The Problem-Independent Mapping
		This can be done with a unitary matrix as close as possible to the nonunitary ideal map to supersets, and that also has a relatively simple form.
		has orthonormal columns and is the closest set of orthogonal vectors according to the Frobenius matrix norm.
		While this gives a set of orthonormal vectors close to the original map, one might be concerned about the requirement to compute the SVD of exponentially large matrices.
	www.cs.cmu.edu /afs/cs.cmu.edu/project/jair/pub/volume4/hogg96a-html/node11.html (1122 words)

	Optimizing Matrix Factorizations for Real-Time Applications (Site not responding. Last check: 2007-10-21)
		The samples could be used to explicitly form a covariance matrix, but moving to the power domain worsens the condition of the matrix, and hence many prefer to perform the QR factorization on the measured (voltage domain) signals.
		For the complex single precision QR matrix factorization, bytes per flop requirements of critical inner loop functions are given in Table 3.
		A typical size of interest for a space-time adaptive processing application might be a matrix with 50 columns (the channels) and 150 rows (the time samples of the interfering clutter and jamming to be canceled).
	www.dtc.army.mil /hpcw/1999/lindermn/index.html (3055 words)

	ORTVEC Call
		For stepwise QR decomposition of a matrix, v is the (n+1)
		Assuming that the m ×n matrix Q has n (nearly) orthonormal columns, the ORTVEC subroutine orthogonalizes the vector v to the columns of Q.
		The case m < n is not possible since Q is assumed to have n (nearly) orthonormal columns.
	www.sfu.ca /sasdoc/sashtml/iml/chap17/sect176.htm (844 words)

	Cameras in Flash
		Multiplying a matrix with a scalar, multiplying a matrix with a vector, and multiplying two matrices.
		To show what an orthonormal matrix is we are going to first associate each row of a matrix with a vector.
		Therefore an orthonormal matrix also means that the vectors a, b, and c are unit vectors and perpendicular.
	members.shaw.ca /mathematica/ahabTutorials/camera.htm (5896 words)

	Converting to and from a view matrix. - MSDN Forums
		The only problem is that the x coordinate of the vectors I transform with the rotation matrix is negated for some reason (not the y and z).
		The problem is that I would actually rather want the rotation in a x, y and z rotation (isn't that called euler angles?) because I believe this will be a lot easier to use in a level editor than quaternions.
		Decomposing a matrix or quaternion into euler angles is possible, but not exactly trivial.
	forums.microsoft.com /MSDN/ShowPost.aspx?PostID=914613&SiteID=1 (819 words)

	Solution to matrix creep. - GameDev.Net Discussion Forums
		I have read somewhere (where I forgot) that making the matrix orthogonal will be the solution, but the algorithm to do this (Gram-Schmidt) doesn't seem very good due to it being iterative, and I'm looking for something fast.
		Posted - 11/6/2005 12:16:41 PM An interesting method is Eric Raible's Matrix Orthogonalization from Graphics Gems I. His methods iterates and finds a new orthogonal ("orthonormal") matrix close to the original matrix.
		Converting a distorted matrix to a quaternion and back to a matrix can also work OK (though for physics applications, it's best to use quaternions and convert once to matrix for point transformations).
	www.gamedev.net /community/forums/topic.asp?topic_id=356162 (1311 words)

	Blender.org :: View topic - scaling along global axis doesnt work?
		Also, this causes the object's orientation matrix to no longer be orthonormal and I don't think openGL would like that.
		I know there are 2 kind of views, 1st which shows distant objects smaller than closer objects, the other shows all objects the same size regardless of their positions.
		Orthonormal means that the vectors defining the matrix' space are all perpendicular to one another.
	www.blender.org /forum/viewtopic.php?p=35929 (1043 words)

	6. Decompositions (Site not responding. Last check: 2007-10-21)
		is the diagonal matrix the diagonal elements of which are the eigenvalues, and P is the matrix obtained by the concatenation of the eigenvectors.
		B associated with the nonzero eigenvalues, and L is a r x r diagonal matrix.
		The LU decomposition of an n-dimensional square matrix A is defined as
	www.cita.nus.edu.sg /mirrors/www.xplore-stat.de/tutorials/matrixnode7.html (774 words)

	IX. EIGENVALUE PROBLEMS
		: For a matrix of the form: b_0 a_0 0...
		Then, the eigenvalues of the nxn matrix M are given by: P_n(z) = 0 (9.3) [Goertzel 1960], p.
		Using these, the definition (9.13) of normalized Hermite polynomials, and Theorem 9.5 provides an expression for the RHS of equation (9.15a) for the matrix elements of XI(n).
	graham.main.nc.us /~bhammel/FCCR/IX.html (2181 words)

	Patent 4772946: Method for data reduction of digital picture signals by vector quantization of coefficients acquired by ...
		The object of the present invention is to provide a method of data reduction wherein the various disadvantages which affect the prior art are avoided In particular, a method is provided for which the need for multiplication operations is eliminated, and which requires a relatively small memory outlay and a relatively short processing time.
		These objects are achieved by a method of data reduction using a Hadamard matrix, in which input vectors are scalarly quantized, and a difference value is developed relative to immediately preceding data, the difference is normalized and vector quantized, and the quantized coefficients are coded and stored as transmitted.
		Various additions and modifications may be made in the method of the present invention without departing from the essential features of novelty thereof, which are intended to be defined and secured by the appended claims.
	www.freepatentsonline.com /4772946.html (2531 words)

	Orthonormal matrix - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		In linear algebra, an orthonormal matrix is a (not necessarily square) matrix with real or complex entries whose columns, treated as vectors in R
		The real n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from an orthogonal matrix; the complex n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from a unitary matrix.
		This page was last modified 22:10, 2 February 2005.
	en.wikipedia.org /wiki/Orthonormal_matrix (141 words)

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