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K-frame
In linear algebra, a ''k''-frame is an ordered set of ''k'' linearly independent vectors in a vector space; thus, ''k'' ≤ ''n'', where ''n'' is the dimension of the space, and an ''n''-frame is precisely an ordered basis. If the vectors are orthogonal, or orthonormal, the frame is called an orthogonal frame, or orthonormal frame, respectively. Properties * The set of ''k''-frames (particularly the set of orthonormal ''k''-frames) in a given vector space ''X'' is known as the Stiefel manifold, and denoted ''V''''k''(''X''). * A ''k''-frame defines a parallelotope (a generalized parallelepiped); the volume can be computed via the Gram determinant. See also * Frame (linear algebra) * Frame of a vector space Riemannian geometry * Orthonormal frame * Moving frame In mathematics, a moving frame is a flexible generalization of the notion of a coordinate frame (an ordered basis of a vector space, in conjunction with an origin) often used to study the extri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stiefel Manifold
In mathematics, the Stiefel manifold V_k(\R^n) is the set of all orthonormal ''k''-frames in \R^n. That is, it is the set of ordered orthonormal ''k''-tuples of vectors in \R^n. It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold V_k(\Complex^n) of orthonormal ''k''-frames in \Complex^n and the quaternionic Stiefel manifold V_k(\mathbb^n) of orthonormal ''k''-frames in \mathbb^n. More generally, the construction applies to any real, complex, or quaternionic inner product space. In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent ''k''-frames in \R^n, \Complex^n, or \mathbb^n; this is homotopy equivalent to the more restrictive definition, as the compact Stiefel manifold is a deformation retract of the non-compact one, by employing the Gram–Schmidt process. Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal Frame
In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If ''M'' is a manifold equipped with a metric ''g'', then an orthonormal frame at a point ''P'' of ''M'' is an ordered basis of the tangent space at ''P'' consisting of vectors which are orthonormal with respect to the bilinear form ''g''''P''.. See also *Frame (linear algebra) *Frame bundle * ''k''-frame *Moving frame *Frame fields in general relativity In general relativity, a frame field (also called a tetrad or vierbein) is a set of four pointwise-orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetim ... References Riemannian geometry {{Riemannian-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frame Of A Vector Space
In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology of signal processing, a frame provides a redundant, stable way of representing a signal. Frames are used in error detection and correction and the design and analysis of filter banks and more generally in applied mathematics, computer science, and engineering. History Because of the various mathematical components surrounding frames, frame theory has roots in harmonic and functional analysis, operator theory, linear algebra, and matrix theory. The Fourier transform has been used for over a century as a way of decomposing and expanding signals. However, the Fourier transform masks key information regarding the moment of emission and the duration of a signal. In 1946, Dennis Gabor was able to solve this using a technique that simultaneously reduced noise, provided resiliency, and created quantization while encapsulatin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. Three equivalent definitions of ''parallelepiped'' are *a hexahedron with three pairs of parallel faces, *a polyhedron with six faces (hexahedron), each of which is a parallelogram, and *a prism (geometry), prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all special cases of parallelepiped. "Parallelepiped" is now usually pronounced or ; traditionally it was because of its etymology in Ancient Greek, Greek παραλληλεπίπεδον ''parallelepipedon'' (with short -i-), a body "having parallel planes". Parallelepipeds are a subclass of the prismatoids. Properties Any of the three pairs of parallel faces can be viewed as the bas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gram Determinant
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\rangle., p.441, Theorem 7.2.10 If the vectors v_1,\dots, v_n are the columns of matrix X then the Gram matrix is X^\dagger X in the general case that the vector coordinates are complex numbers, which simplifies to X^\top X for the case that the vector coordinates are real numbers. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the #Gram determinant, Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram. Examples For finite-dimensional real vectors in \mathbb^n with the usual Euclidean dot product, the Gram matrix is G = V^\top V, where V is a matrix whose columns are the vectors v_k and V^\top is its transpose whose r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Set
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However some a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormality
In linear algebra, two vector space, vectors in an inner product space are orthonormal if they are orthogonality, orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis (linear algebra), basis is called an ''orthonormal basis''. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian coordinate system#Cartesian coordinates in two dimensions, Cartesian plane, two Vector (geometry), vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically used for lines and planes that intersect to form a right angle, whereas ''orthogonal'' is used in generalizations, such as ''orthogonal vectors'' or ''orthogonal curves''. ''Orthogonality'' is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics Optics In optics, polarization states are said to be ort ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
