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Linear algebra operation
In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product
Orthogonal_transformation
Real square matrix whose columns and rows are orthogonal unit vectors
real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product
Orthogonal_matrix
Mathematical transformation that preserves distances
represents a rotation (an orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant
Rigid_transformation
Concept in physics and mathematics
→ R3 is an orthogonal transformation. As a Lie group, the group of Galilean transformations has dimension 10. Two Galilean transformations G(R, v, a,
Galilean_transformation
Specific linear basis (mathematics)
of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for R n {\displaystyle
Orthonormal_basis
Endomorphism preserving the inner product
Antiunitary Orthogonal transformation Time reversal Unitary group Unitary operator Unitary matrix Wigner's theorem Unitary transformations in quantum mechanics
Unitary_transformation
Rational function of the form (az + b)/(cz + d)
of translations, similarities, orthogonal transformations and inversions. The general form of a Möbius transformation is given by f ( z ) = a z + b c
Möbius_transformation
of an orthogonal transformation (an origin-preserving rigid transformation) with a uniform scaling (dilation). All similarity transformations (which
Conformal linear transformation
Conformal_linear_transformation
Vector space with generalized dot product
factors and orthogonal directions of scaling. It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation. The article
Inner_product_space
Mathematic theorem
after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described
Cartan–Dieudonné_theorem
Idempotent linear transformation from a vector space to itself
Projections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems: QR decomposition (see Householder transformation and
Projection_(linear_algebra)
Matrix with nonzero elements on the main diagonal and the diagonals above and below it
tridiagonal matrix is given by the continuant of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form can be
Tridiagonal_matrix
Matrix decomposition
The matrix Q is orthogonal and R is upper triangular, so A = QR is the required QR decomposition. The use of Householder transformations is inherently the
QR_decomposition
Relation of space and time in relativity theory
pair of conjugate hyperbolas, two conjugate diameters are hyperbolically orthogonal. This relationship of diameters was described by Apollonius of Perga and
Hyperbolic_orthogonality
to a given orthogonal transformation ( A {\displaystyle {\boldsymbol {A}}} ) if it does not change when subjected to that transformation. For invariance
Orthotropic_material
Type of group in mathematics
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension
Orthogonal_group
Generalization of perpendicularity
linear transformation preserves angles and distance ratios, meaning that transforming orthogonal vectors by the same conformal linear transformation will
Orthogonality_(mathematics)
Central object in linear algebra; mapping vectors to vectors
with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections
Transformation_matrix
Topological space in group theory
under an orthogonal transformation from O(n − 1). This shows us why we can construct Sn−1 as a homogeneous space. Oriented sphere (special orthogonal group):
Homogeneous_space
Method of data analysis
used to interpret findings of the PCA. PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a
Principal_component_analysis
Theorem limiting types of conformal mappings in Euclidean space of dimension > 2
composition of translations, homotheties, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions). This theorem severely
Liouville's theorem (conformal mappings)
Liouville's_theorem_(conformal_mappings)
Isometry group of Euclidean space
{\displaystyle x\mapsto A(x+b)} where A is an orthogonal matrix or the same orthogonal transformation followed by a translation: x ↦ A x + c , {\displaystyle
Euclidean_group
Random matrix with gaussian entries
properties: Invariance under orthogonal transformation: For any fixed (not random) N × N {\displaystyle N\times N} orthogonal matrix O {\displaystyle O}
Gaussian_ensemble
Geometric symmetry operation
Cartesian coordinate system. Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by − 1 {\displaystyle -1}
Point_reflection
Representation of a tensor in Euclidean space
components from one such basis to another is done through an orthogonal transformation. The most familiar coordinate systems are the two-dimensional
Cartesian_tensor
Algebra based on a vector space with a quadratic form
is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of
Clifford_algebra
Matrix of inner products of vectors
rotation or reflection of R k {\displaystyle \mathbb {R} ^{k}} (any orthogonal transformation, that is, any Euclidean isometry preserving 0) to the sequence
Gram_matrix
Family of linear transformations
on spacetime, and the group of transformations which leaves this quadratic form invariant is the indefinite orthogonal group O(3,1), a Lie group. In other
Lorentz_transformation
Means of projecting three-dimensional objects in two dimensions
projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface
Orthographic_projection
Concept in linear algebra
) that is orthogonal to the hyperplane. The reflection of a point x {\textstyle x} about this hyperplane is the Householder transformation: x → − 2 ⟨
Householder_transformation
Geometry founded on spheres
general Lie transformations. The subgroup of Lie transformations preserving the point cycles is essentially the subgroup of orthogonal transformations which
Lie_sphere_geometry
Property that is not changed by mathematical transformations
invariant under orthogonal transformations. Area is invariant under linear maps which have determinant ±1 (see Equiareal map § Linear transformations). Some invariants
Invariant_(mathematics)
Geological concept
to a given orthogonal transformation ( A {\displaystyle {\boldsymbol {A}}} ) if it does not change when subjected to that transformation. For invariance
Transverse_isotropy
Coordinate system using perpendicular axes
the affine transformations, the Euclidean transformations are characterized by the fact that the matrix A {\displaystyle A} is orthogonal; that is, its
Cartesian_coordinate_system
Measure of covariance of components of a random vector
symmetric, it can be diagonalized by a linear orthogonal transformation, i.e. there exists such orthogonal matrix A {\displaystyle \mathbf {A} } (meanwhile
Covariance_matrix
Form of a matrix
every skew-symmetric matrix to a block diagonal form by a special orthogonal transformation. Specifically, every 2 n × 2 n {\displaystyle 2n\times 2n} real
Skew-symmetric_matrix
Algebraic structure designed for geometry
allows the modeling of Euclidean transformations of R 3 {\displaystyle \mathbb {R} ^{3}} as orthogonal transformations of a subset of R 4 , 1 {\displaystyle
Geometric_algebra
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
k-frame, and any two k-frames are related by some orthogonal transformation. In other words, the orthogonal group O(n) acts transitively on V k ( R n ) .
Stiefel_manifold
Antisymmetric permutation object acting on tensors
systems related by orthogonal transformations. However, the Levi-Civita symbol is a pseudotensor because under an orthogonal transformation of Jacobian determinant
Levi-Civita_symbol
Spin representations of the SO(3) group
of two reflections. (More generally, any orientation-reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus
Spinors_in_three_dimensions
Geometric transformation that preserves lines but not angles nor the origin
similarity transformations form the subgroup where A {\displaystyle A} is a scalar times an orthogonal matrix. For example, if the affine transformation acts
Affine_transformation
Method for specifying point positions
intersection of curves. Orthogonal coordinates: coordinate surfaces meet at right angles Skew coordinates: coordinate surfaces are not orthogonal The log-polar
Coordinate_system
Mathematical problem
combinatorics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Type of signal in signal processing
has spherical symmetry in n-dimensional space. Therefore, any orthogonal transformation of the vector will result in a Gaussian white random vector. In
White_noise
Matrix representing a Euclidean rotation
matrix inversion (for these orthogonal matrices equivalently matrix transpose). Alias or alibi (passive or active) transformation The coordinates of a point
Rotation_matrix
Tensor that rotates the reference frame to simplify analysis
will be orthogonal to the plane of the two-dimensional perspective mentioned above. The first step towards building the Clarke transformation requires
Direct-quadrature-zero transformation
Direct-quadrature-zero_transformation
Development of linear transformations forming the Lorentz group
(1904/05). The Wikiversity: History of Lorentz transformations via imaginary orthogonal transformation includes contributions of Sophus Lie (1871), Hermann
History of Lorentz transformations
History_of_Lorentz_transformations
Scalar quantity, changing sign in mirrored coordinates
e_{2})\mapsto (u_{1},u_{2})} is a change of basis representing an orthogonal transformation, then e 1 e 2 ↦ u 1 u 2 = ± e 1 e 2 , {\displaystyle e_{1}e_{2}\mapsto
Pseudoscalar
Assignment of a vector to each point in a subset of Euclidean space
\mathbb {R} ))} where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0. The point 0 is called
Vector_field
Engineering model
with respect to monotonic transformations of the function (scaling) Invariance with respect to orthogonal transformations of the search space (rotation)
Surrogate_model
Characteristic of state of a computer system that outlives the process that created it
to be "orthogonal" or "transparent" when it is implemented as an intrinsic property of the execution environment of a program. An orthogonal persistence
Persistence (computer science)
Persistence_(computer_science)
notation "Z" is because the scalar transformations are the center of the orthogonal group. The projective special orthogonal group, PSO, is defined analogously
Projective_orthogonal_group
Mathematical function that preserves angles
coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with
Conformal_map
Matrix decomposition
orthogonal m × m {\displaystyle m\times m} matrices. M {\displaystyle \mathbf {M} } can be interpreted to represent a linear transformation
Singular_value_decomposition
Concept in mathematical group theory
orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations
Conformal_group
Geometrical property
are possible in geometry. By the Cartan–Dieudonné theorem, an orthogonal transformation in n-dimensional space can be represented by the composition of
Symmetry_(geometry)
Non-tensorial representation of the spin group
g), i.e., V is equipped with a complex structure J that is an orthogonal transformation with respect to the inner product g on V. Then V ⊗ R C {\displaystyle
Spinor
Subgroup of the Clifford algebra associated to a quadratic space
the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the orthogonal group
Pin_group
,\phi ^{n})} , then the two coframes would be related by an orthogonal transformation Φ = M Θ {\displaystyle \Phi =M\Theta } If the connection 1-form
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
In mathematics, invariant of square matrices
determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by
Determinant
Statistical method
a set of factors and factor loadings is unique only up to an orthogonal transformation. Suppose a psychologist has the hypothesis that there are two
Factor_analysis
Computer language used to make queries into databases and information systems
language for transforming data. Consists of a curated set of orthogonal transformations, which are combined together to form a pipeline. PTQL based on
Query_language
Concept in linear algebra
the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle
Orthogonal_complement
Concepts from linear algebra
reversed) by a given linear transformation. More precisely, an eigenvector v {\displaystyle \mathbf {v} } of a linear transformation T {\displaystyle T} is
Eigenvalues_and_eigenvectors
{\omega }}} is the following. Because Ω is the derivative of an orthogonal transformation, the bilinear form B ( r , s ) = ( Ω r ) ⋅ s {\displaystyle B(\mathbf
Angular_velocity_tensor
Mathematical concept
interferometers (corresponding to orthogonal matrices O and O' ) intermitted by a layer of active non-linear squeezing transformations (given in terms of the matrix
Symplectic_matrix
Physical theory with fields invariant under the action of local "gauge" Lie groups
the transformation Φ ↦ Φ ′ = G Φ {\displaystyle \ \Phi \mapsto \Phi '=G\Phi } whenever G is a constant matrix belonging to the n-by-n orthogonal group
Gauge_theory
Mathematical transformation
In mathematics, the Stieltjes transformation Sρ(z) of a measure of density ρ on a real interval I is the function of the complex variable z defined outside
Stieltjes_transformation
Mathematical method
Alberto; Haine, Luc (1996). "Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation". Symmetries and Integrability
Darboux_transformation
Natural moving frame in differential geometry of surfaces
) = A x + x 0 {\displaystyle \phi (x)=Ax+x_{0}} where A is an orthogonal transformation and x0 is a translation. Then, on a frame, ϕ ( v ; f 1 , … , f
Darboux_frame
Matrix approximation problem in linear algebra
The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle
Orthogonal_Procrustes_problem
Matrix realization of the Clifford algebra
now turn to the action of the orthogonal group on the spinors. Consider the application of an orthogonal transformation to the coordinates, which in turn
Weyl–Brauer_matrices
Stress-strain relation in a linear elastic material
of transformations, formally known as a group operation. For example, an invariant with respect to the group of proper orthogonal transformations, called
Elasticity_tensor
Fundamental space of geometry
subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect
Euclidean_space
Geometrical figure in a Euclidean space
their relationship with the geometry of polytopes and groups of orthogonal transformations. Schläfli showed early on that the vectors from the center of
Eutactic_star
Mathematics concept
is an orthogonal transformation. Likewise, J preserves a nondegenerate, skew-symmetric form ω if and only if J is a symplectic transformation (that is
Linear_complex_structure
Motion of a certain space that preserves at least one point
orthogonal matrix must be 1. The only other possibility for the determinant of an orthogonal matrix is −1, and this result means the transformation is
Rotation_(mathematics)
2D modulation technique
Orthogonal Time Frequency Space (OTFS) is a 2D modulation technique that transforms the information carried in the Delay-Doppler coordinate system. The
Orthogonal Time Frequency Space
Orthogonal_Time_Frequency_Space
Mathematical transformation
function f is symmetric with respect to a given set G of orthogonal linear transformations, f ( A x ) = f ( x ) , ∀ x , ∀ A ∈ G {\displaystyle f(Ax)=f(x)
Legendre_transformation
Feature of a system that is preserved under some transformation
that is preserved or remains unchanged under some transformation. A family of particular transformations may be continuous (such as rotation of a circle)
Symmetry_(physics)
Geometric operation
cutting them off. Procrustes analysis Orthogonal Procrustes problem Singular value decomposition Affine transformation, which also allows for shear Ramsay
Procrustes_transformation
Orthogonal group of an indefinite quadratic form
the indefinite orthogonal group, O ( p , q ) {\displaystyle \operatorname {O} (p,q)} is the Lie group of all linear transformations of an n {\displaystyle
Indefinite_orthogonal_group
Group of geometric symmetries with at least one fixed point
are subgroups of the special orthogonal group SO(d): they contain only orientation-preserving orthogonal transformations, i.e., those of determinant +1
Point_group
Differential operator in mathematics
f)\circ g.} Thus the Laplacian commutes with translations and with orthogonal transformations, hence in particular with rotations and reflections. In two dimensions
Laplace_operator
Branch of mathematics
in such a position, and the linear bijection is unique up to orthogonal transformations. A convex body is in the John position if its maximal volume inscribed
Asymptotic_geometry
Type of matrix
possible via rigid transformations, usually using singular value decomposition. The ordinary Euclidean case is known as the orthogonal Procrustes problem
Euclidean_distance_matrix
Procedure to convert 3D scenes to 2D images
visual volume is therefore a truncated pyramid (frustum). The parallel or orthogonal projection is used, for example, for technical representations because
Graphics_pipeline
)}(R\mathbf {y} )=Z_{\mathbf {x} }^{(\ell )}(\mathbf {y} )} for every orthogonal transformation R. Conversely, any function f(x,y) on Sn−1×Sn−1 that is a spherical
Zonal_spherical_harmonics
Coordinate system whose directions vary in space
coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r. The orthogonal basis vectors are br = (cos θ, sin θ), bθ =
Curvilinear_coordinates
Coordinate transformation that preserves the form of Hamilton's equations
are not the same as a physical rotation of an orthogonal spatial coordinate system. The transformation ( Q ( q , p ) , P ( q , p ) ) = ( q + f ( p )
Canonical_transformation
Approximation method in statistics
method of orthogonal decomposition involves singular value decomposition, in which R is diagonalized by further orthogonal transformations. J = U Σ V
Non-linear_least_squares
Concept involving selective reactions
groups. Orthogonal protection is widely used in organic chemistry and synthetic chemistry to allow chemists to perform multiple transformations on complex
Orthogonality_(chemistry)
sub-group of S O ( 4 n ) {\displaystyle SO(4n)} consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic n × n {\displaystyle
Quaternion-Kähler_manifold
Set of coordinates where the coordinate hypersurfaces all meet at right angles
In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots
Orthogonal_coordinates
French banker and mathematician (1795–1851)
Cayley acknowledged Euler's and Rodrigues' priority describing orthogonal transformations. Rodrigues is credited as originating the idea of the artist as
Olinde_Rodrigues
Linear map that preserves areas
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves the Euclidean area of regions in the
Squeeze_mapping
Process in linear algebra
linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly
Orthogonalization
Mathematical theory
Killing form. Thus relative to this inner product, Ad(G) acts by orthogonal transformations ( SO ( g ) {\displaystyle \operatorname {SO} ({\mathfrak {g}})}
Compact_Lie_algebra
Isomorphism of projective spaces in geometry
of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation. Historically, homographies (and projective
Homography
Polynomial sequence
mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate
Zernike_polynomials
ORTHOGONAL TRANSFORMATION
ORTHOGONAL TRANSFORMATION
ORTHOGONAL TRANSFORMATION
ORTHOGONAL TRANSFORMATION
ORTHOGONAL TRANSFORMATION
ORTHOGONAL TRANSFORMATION
ORTHOGONAL TRANSFORMATION
ORTHOGONAL TRANSFORMATION
ORTHOGONAL TRANSFORMATION