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Topic: Orthogonal matrix

Related Topics

Inverse matrix

Matrix (mathematics)

Unitary matrix

Normal matrix

Hermitian matrix

Matrix inversion

Symmetric matrix

Matrix exponential

Householder reflection

QR decomposition

Hermitian conjugate

Conjugate transpose

QR algorithm

Givens rotation

Whitening

	Orthogonal matrix - Wikipedia, the free encyclopedia
		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.
		The rest of the matrix is an n×n orthogonal matrix; thus O(n) is a subgroup of O(n+1) (and of all higher groups).
		The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular.
	en.wikipedia.org /wiki/Orthogonal_matrix (2819 words)

	Matrix (mathematics) - Wikipedia, the free encyclopedia
		The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A.
		A 1 × n matrix (one row and n columns) is called a row vector, and an m × 1 matrix (one column and m rows) is called a column vector.
		The rank of a matrix A is the dimension of the image of the linear map represented by A; this is the same as the dimension of the space generated by the rows of A, and also the same as the dimension of the space generated by the columns of A.
	en.wikipedia.org /wiki/Matrix_(mathematics) (1668 words)

	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 (1452 words)

	PlanetMath: orthogonal matrices (Site not responding. Last check: 2007-11-07)
		Orthogonal matrices can be thought of as the real case of unitary matrices.
		The characteristic polynomial of a orthogonal matrix is a reciprocal polynomial
		This is version 6 of orthogonal matrices, born on 2002-01-02, modified 2003-03-31.
	planetmath.org /encyclopedia/OrthogonalMatrices.html (225 words)

	GameDev.net - Quaternions and Orthogonal 4x4 Real Matrices
		The _inverse_ of a matrix M (assuming that it has one, i.e., det(M)/=0) is M^-1 -- i.e, M M^-1 = M^-1 M = I, where I is the _identity_ matrix of the same size as M. The inverse of [W X] [ Z -X] M = [Y Z] is (1/det(M)) [-Y W].
		Trivial properties of orthogonal matrices: * the transpose and inverse of an orthogonal matrix are themselves orthogonal.
		The _norm_ of a quaternion Q(a,b,c,d) is the square root of the _determinant_ of the corresponding matrix: det(Q(a,b,c,d)) = (a^2 + b^2 + c^2 + d^2)^2 norm(Q(a,b,c,d)) = a^2 + b^2 + c^2 + d^2.
	www.gamedev.net /reference/articles/article428.asp (1849 words)

	PlanetMath: Cayley's parameterization of orthogonal matrices (Site not responding. Last check: 2007-11-07)
		In particular, it is useful because unitary and orthogonal operators are bounded whereas Hermitian and skew-symmetric operators may or may not be bounded.
		For instance, it is often easier to obtain the spectral decomposition of a Hermitian operator or study symmetric extensions of a symmetric operator by first performing a Cayley transform and dealing with the resulting bounded operator.
		This is version 12 of Cayley's parameterization of orthogonal matrices, born on 2004-11-27, modified 2005-02-19.
	planetmath.org /encyclopedia/CayleysParameterizationOfOrthogonalMatrices.html (574 words)

	Matrix Reference Manual: Special Matrices
		A 2#2 orthogonal matrix is either a Givens rotation or a Givens reflection according to whether it is proper or improper.
		A 3#3 orthogonal matrix is either a rotation matrix or else a rotation matrix plus a reflection in the plane of the rotation according to whether it is proper or improper.
		A Rotation matrix is orthogonal with a determinant of +1.
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/special.html (4418 words)

	Symmetric matrix: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-07)
		In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure....
		In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix a whose transpose is also its negative; that is, it satisfies...
		In linear algebra, a circulant matrix is a special kind of toeplitz matrix where each row vector is shifted one element to the right relative to the preceding...
	www.absoluteastronomy.com /encyclopedia/s/sy/symmetric_matrix.htm (834 words)

	PlanetMath: geometry (Site not responding. Last check: 2007-11-07)
		Early differential geometers studied such properties of curves and surfaces such as: computing their lengths and areas, finding tangents, constructing evolute, involute, and pedal curves, studying curvature and osculating circles, and finding envelopes and orthogonal curves to a given family of curves.
		In the case of a surface, the situation is a little more complicated -- to describe the direction of the normal vector, one needs two angles instead of one and one can choose to compute their directional derivatives along any direction tangent to the surface.
		Hence, to define the notion of metric field, we will consider the assignment of a symmetric, positive definite matrix of functions to every coordinate system in such a way that the matrices assigned to two coordinate systems are related according to the transformation law for inner products under a change of basis.
	planetmath.org /encyclopedia/Geometry.html (4214 words)

	Linear Algebra (Math 2318) - Vector Spaces - Orthogonal Matrices (Site not responding. Last check: 2007-11-07)
		As noted above, in order for a matrix to be an orthogonal matrix it must be square. So a matrix that is not square, but does have orthonormal columns will not be orthogonal. Also, note that we did mean to say that the columns are orthonormal.
		and by construction this matrix has orthonormal columns and since it is a square matrix it is an orthogonal matrix.
		Again, by construction this matrix has orthonormal columns. However, since it is not a square matrix it is NOT an orthogonal matrix.
	tutorial.math.lamar.edu /AllBrowsers/2318/OrthogonalMatrix.asp (962 words)

	Matrix Manual: Special Matrices (Site not responding. Last check: 2007-11-07)
		matrix, A, is an nn Toeplitz matrix* in which a(i,j) is a function of {(i-j) modulo n}.
		A is symmetric iff it is congruent to a diagonal matrix.
		Any square matrix may be uniquely decomposed as the sum of a symmetric matrix and a skew-symmetric matrix.
	www.psi.toronto.edu /matrix/special.html (2192 words)

	Inverting an Orthogonal Matrix (Site not responding. Last check: 2007-11-07)
		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)

	MATH2071: LAB #10: QR Factorizations (Site not responding. Last check: 2007-11-07)
		By convention, an orthogonal matrix is usually denoted by the symbol Q.
		the identity matrix is orthogonal; the matrix [0 1; 1 0] is orthogonal; the matrix [c s; -s c] is orthogonal, where c and s are the cosine and sine of some angle; any diagonal matrix is orthogonal.
		Conversely, if a matrix has columns of unit L2 norm which are pairwise orthogonal...it may loosely be called an orthogonal matrix, but it isn't, unless the matrix is square.
	www.csit.fsu.edu /~burkardt/math2071/lab_10.html (3230 words)

	ipedia.com: Orthogonal group Article (Site not responding. Last check: 2007-11-07)
		In mathematics, the orthogonal group of degree n over a field F) is the group of n -by- n orthogonal matrices with entries from F, with the group operation that of matrix multiplication.
		Every orthogonal matrix has determinant either 1 or -1.
		Both O(n,F) and SO(n,F) are algebraic groups, because the condition that a matrix have its own transpose as inverse can be expressed as a set of polynomial equations in the entries of the matrix.
	www.ipedia.com /orthogonal_group.html (610 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-11-07)
		A border banded matrix is a 2 by 2 block matrix comprising a (large) leading block which is a square banded matrix, two dense rectangular side strips, and a (small) trailing block which is a square dense matrix.
		The inverse of a (nonsingular) circulant matrix is a circulant matrix.
		The transpose of a circulant matrix is a circulant matrix.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13644 words)

	Tests for the Orthogonal Factorization Routines
		The subroutines to generate or multiply by the orthogonal matrix from the factorization are also tested in these paths.
		There is not a separate test path for the orthogonal transformation routines, since the important thing when generating an orthogonal matrix is not whether or not it is, in fact, orthogonal, but whether or not it is the orthogonal matrix we wanted.
		To obtain an upper trapezoidal matrix with the specified singular value distribution, we generate a dense matrix using SLATMS and reduce it to upper triangular form using SGEQR2.
	www.netlib.org /lapack/lawn41/node36.html (591 words)

	COMPORT Call
		If b is not specified, q is the m ×m orthogonal matrix Q that is the product of the min(m,n) separate Householder transformations.
		If b is specified, q is the m ×p matrix Q' B that has the transposed Householder transformations Q' applied on the p columns of the argument matrix B.
		specifies an optional m ×p matrix B that is to be left multiplied by the transposed m ×m matrix Q'.
	www.asu.edu /it/fyi/unix/helpdocs/statistics/sas/sasdoc/sashtml/iml/chap17/sect42.htm (425 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		As the page above specifies this means that if you create a set of orthogonal vectors A,B,C general rotations are very easy to do, and goes on to provide a two part solution to rotating any general vector to any other general vector (while retaining an general up direction).
		Given a orthogonal rotation matrix Then the axis and angle of rotation can be found using...
		We need to find two more unit vectors orthogonal to the original vector V. One of these vectors must be the axis of rotation so it can be moved to the Z axis for the second step.
	www.cit.gu.edu.au /~anthony/info/graphics/matrix.hints (685 words)

	Symmetric Matrices
		This says that a symmetric matrix with distinct roots is always similar to a diagonal matrix.
		There is special property that holds for orthogonal matrices that is worth noting.
		This follows immediately from the definition of orthogonal and matrix multiplication.
	www.ltcconline.net /greenl/courses/203/MatrixOnVectors/symmetricMatrices.htm (232 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)

	Assumed knowledge of matrices and vector spaces
		The transpose of an orthogonal matrix is also orthogonal.
		The eigenvectors corresponding to two different eigenvalues are orthogonal.
		is a diagonal matrix whose elements are the eigenvalues of
	mcs.une.edu.au /~stat354/notes/node13.html (244 words)

	Unitary and Orthogonal Transforms
		) of a unitary or orthogonal matrix are orthogonal (perpendicular to each other) and normalized (of unit length), or orthonormal, i.e., their inner product satisfies:
		As the n-dimensional space can be spanned by the column vectors of any n by n unitary (orthogonal) matrix, a vector
		As any two orthogonal coordinate systems can be related by a rotation, Fourier transform
	fourier.eng.hmc.edu /e101/lectures/Image_Processing/node15.html (279 words)

	Orthogonal Factorizations and Linear Least Squares Problems (Site not responding. Last check: 2007-11-07)
		Orthogonal or unitary matrices have the important property that they leave the two-norm of a vector invariant:
		Orthogonal factorizations are used in the solution of linear least squares problems .
		Table 3.7 lists all routines provided by ScaLAPACK to perform orthogonal factorizations and the generation or pre- or post-multiplication of the matrix Q for each matrix type and storage scheme.
	www.netlib.org /scalapack/slug/node52.html (142 words)

	M314 Practice Exam 3 Solutions (Site not responding. Last check: 2007-11-07)
		Given an eigenvalue l of a matrix A, recall that its eigenspace is the nullspace of A-lI.
		to get the columns of P; D is the diagonal matrix whose entries are the eignevalues (taken in the same order as we took their eigenvectors as columns in P).
		This is OK since it still gives an orthogonal basis of V. (b) Find the projection of the vector (1,3,1) into the subspace V of part (a).
	www.math.unl.edu /~bharbourne1/M314F05/M314F05PracticeExam3Solns.html (632 words)

	No Title
		How is the determinant of a matrix A related to the determinant of A
		For a given matrix A, how do you decide if a given number is an eigenvalue of A?
		Suppose a square matrix U has orthonormal columns.
	www.math.mtu.edu /~trolson/review2321b/review2321b.html (387 words)

	Section 3
		is called an orthogonal transformation if it preserves the norm of vectors.
		A Householder matrix, sometimes called a Householder reflection, is an
		This technique has advantages in computational problems since mutliplication by orthogonal matrices is numerically stable, where as the Gram-Schmidt technique for finding
	www-math.cudenver.edu /~rrosterm/lins05_8sol/node3.html (119 words)

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