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Uniform Distribution On A Stiefel Manifold
The uniform distribution on a Stiefel manifold is a matrix-variate distribution that plays an important role in multivariate statistics. There one often encounters integrals over the orthogonal group or over the Stiefel manifold with respect to an invariant measure. For example, this distribution arises in the study of the functional determinant under transformations involving orthogonal or semi-orthogonal matrices. The uniform distribution on the Stiefel manifold corresponds to the normalized Haar measure on the Stiefel manifold. A random matrix uniformly distributed on the Stiefel manifold is invariant under the two-sided group action of the product O(p)\times O(n) of orthogonal groups, i.e. X \sim V_1 X V_2 for all V_1 \in O(p) and V_2 \in O(n). Uniform Distribution on a Stiefel Manifold Introduction Let V_ := V_n(\mathbb^p) be the Stiefel manifold, i.e., the set of all orthonormal n-frames in \mathbb^p for n \leq p. This manifold can also be represented as the matrix set ... [...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] |
Multivariate Statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable, i.e., '' multivariate random variables''. Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied. In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both :*how these can be used to represent the distributions of observed data; :*how they can be used as part of statistical inference, particularly where several different quantities are of interest to the same analysis. Certain types of problems involving multivariate da ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by Function composition, composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrix, orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose invertible matrix, inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact group, compact. The orthogonal group in dimension has two connected component (topology), connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Determinant
In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the infinite-dimensional case of a linear operator ''S'' mapping a function space ''V'' to itself. The corresponding quantity det(''S'') is called the functional determinant of ''S''. There are several formulas for the functional determinant. They are all based on the fact that the determinant of a finite matrix is equal to the product of the eigenvalues of the matrix. A mathematically rigorous definition is via the zeta function of the operator, : \zeta_S(a) = \operatorname\, S^ \,, where tr stands for the functional trace: the determinant is then defined by : \det S = e^ \,, where the zeta function in the point ''s'' = 0 is defined by analytic continuation. Another possible generalization, often used by physicists when using the Feynman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of mathematical analysis, analysis, number theory, group theory, representation theory, mathematical statistics, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact space, locally compact Hausdorff space, Hausdorff topological group. The Sigma-algebra, \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right Coset, translates of S by ''g'' as follows: * Left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the study of properties of random matrices, often as they become large. RMT provides techniques like mean-field theory, diagrammatic methods, the cavity method, or the replica method to compute quantities like traces, spectral densities, or scalar products between eigenvectors. Many physical phenomena, such as the spectrum of nuclei of heavy atoms, the thermal conductivity of a lattice, or the emergence of quantum chaos, can be modeled mathematically as problems concerning large, random matrices. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Space
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Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces *Quotient space (linear algebra), in case of vector spaces *Quotient space of an algebraic stack *Quotient metric space See also *Quotient object In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) \, dx is an example of a -form, and can be integrated over an interval ,b/math> contained in the domain of f: \int_a^b f(x)\,dx. Similarly, the expression f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz is a -form that can be integrated over a surface S: \int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right). The symbol \wedge denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form f(x,y,z) \, dx \wedge dy \wedge dz represents a volume element that can be integrated over a region of space. In general, a -form is an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Gamma Function
In mathematics, the multivariate gamma function Γ''p'' is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate beta distribution. It has two equivalent definitions. One is given as the following integral over the p \times p positive-definite real matrices: : \Gamma_p(a)= \int_ \exp\left( -(S)\right)\, \left, S\^ dS, where , S, denotes the determinant of S. The other one, more useful to obtain a numerical result is: : \Gamma_p(a)= \pi^\prod_^p \Gamma(a+(1-j)/2). In both definitions, a is a complex number whose real part satisfies \Re(a) > (p-1)/2. Note that \Gamma_1(a) reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for p\ge 2: : \Gamma_p(a) = \pi^ \Gamma(a) \Gamma_(a-\tfrac) = \pi^ \Gamma_(a) \Gamma(a+(1-p)/2). Thus * \Gamma_2(a)=\pi^\Gamma(a)\G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
