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Gramian
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^* 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 (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 rows are the vectors v_k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Observability Gramian
In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is observable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are, respectively, n\times n, n\times p,q\times n and q\times p matrices. One of the many ways one can achieve such goal is by the use of the Observability Gramian. Observability in LTI Systems Linear Time Invariant (LTI) Systems are those systems in which the parameters \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are invariant with respect to time. One can determine if the LTI system is or is not observable simply by looking at the pair (\boldsymbol,\boldsymbol). Then, we can say that the following statements are equivalent: 1. The pair (\boldsymbol,\boldsymbol) is observable. 2. The n\times n matrix \boldsymbol(t)=\int_^e^\boldsymbol^\boldsymbole^d\tau is nonsingular for any t>0. 3. The nq\times n observability matrix ... [...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^* 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 (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 rows are the vectors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Controllability Gramian
In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are, respectively, n\times n, n\times p, q\times n and q\times p matrices. One of the many ways one can achieve such goal is by the use of the Controllability Gramian. Controllability in LTI Systems Linear Time Invariant (LTI) Systems are those systems in which the parameters \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are invariant with respect to time. One can observe if the LTI system is or is not controllable simply by looking at the pair (\boldsymbol,\boldsymbol). Then, we can say that the following statements are equivalent: 1. The pair (\boldsymbol,\boldsymbol) is controllable. 2. The n\times n matrix \boldsymbol(t)=\int_^e^\boldsymbole^d\tau=\int_^e^\boldsymbole^d\tau is nonsingular for any t>0. 3. The n\times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jørgen Pedersen Gram
Jørgen Pedersen Gram (27 June 1850 – 29 April 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark. Important papers of his include ''On series expansions determined by the methods of least squares'', and ''Investigations of the number of primes less than a given number''. The mathematical method that bears his name, the Gram–Schmidt process, was first published in the former paper, in 1883. For number theorists his main fame is the series for the Riemann zeta function (the leading function in Riemann's exact prime-counting function). Instead of using a series of logarithmic integrals, Gram's function uses logarithm powers and the zeta function of positive integers. It has recently been supplanted by a formula of Ramanujan that uses the Bernoulli numbers directly instead of the zeta function. In control theory, the Gramian or Gram matrix is an important contribution named after him. The C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Principal Component Analysis
In the field of multivariate statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space. Background: Linear PCA Recall that conventional PCA operates on zero-centered data; that is, :\frac\sum_^N \mathbf_i = \mathbf, where \mathbf_i is one of the N multivariate observations. It operates by diagonalizing the covariance matrix, :C=\frac\sum_^N \mathbf_i\mathbf_i^\top in other words, it gives an eigendecomposition of the covariance matrix: :\lambda \mathbf=C\mathbf which can be rewritten as :\lambda \mathbf_i^\top \mathbf=\mathbf_i^\top C\mathbf \quad \textrm~i=1,\ldots,N. (See also: Covariance matrix as a linear operator) Introduction of the Kernel to PCA To understand the utility of kernel PCA, particularly for clustering, observe that, while ''N'' points cannot, in general, be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The simpl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overlap Matrix
In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. Linus Pauling explained the importance of orbital overlap in the molecular bond angles observed through experimentation; it is the basis for orbital hybridization. As ''s'' orbitals are spherical (and have no directionality) and ''p'' orbitals are oriented 90° to each other, a theory was needed to explain why molecules such as methane (CH4) had observed bond angles of 109.5°. Pauling proposed that s and p orbitals on the carbon atom can combine to form hybrids (sp3 in the case of methane) which are directed toward the hydrogen atoms. The carbon hybrid orbitals have greater overlap with the hydrogen orbitals, and can therefore form stronger C–H bonds.Pauling, Linus. (1960). ''The Nature Of The Chemical Bond''. Cornell University Press. A quantitative measure of the overlap of two atomic orbitals ΨA and ΨB ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2 \times 2 matrix would be necessary to fully characterize the two-dimensional variation. The covariance matrix of a random vector \mathbf is typically denoted by \operatorname_ or \Sigma. Definition Throughout this article, boldfaced unsub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Function
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas. This article will discuss some of the historical and current developments of the theory of positive-definite kernels, starting with the general idea and properties before considering practical applications. Definition Let \mathcal X be a nonempty set, sometimes referred to as the index set. A symmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data, known as training data, in order to make predictions or decisions without being explicitly programmed to do so. Machine learning algorithms are used in a wide variety of applications, such as in medicine, email filtering, speech recognition, agriculture, and computer vision, where it is difficult or unfeasible to develop conventional algorithms to perform the needed tasks.Hu, J.; Niu, H.; Carrasco, J.; Lennox, B.; Arvin, F.,Voronoi-Based Multi-Robot Autonomous Exploration in Unknown Environments via Deep Reinforcement Learning IEEE Transactions on Vehicular Technology, 2020. A subset of machine learning is closely related to computational statistics, which focuses on making predicti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systems Theory
Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structure, function and role, and expressed through its relations with other systems. A system is "more than the sum of its parts" by expressing synergy or emergent behavior. Changing one component of a system may affect other components or the whole system. It may be possible to predict these changes in patterns of behavior. For systems that learn and adapt, the growth and the degree of adaptation depend upon how well the system is engaged with its environment and other contexts influencing its organization. Some systems support other systems, maintaining the other system to prevent failure. The goals of systems theory are to model a system's dynamics, constraints, conditions, and relations; and to elucidate principles (such as purpose, measur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
