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Jordan Matrix
In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form: \begin \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & 0 & \lambda \end . Definition Every Jordan block is specified by its dimension ''n'' and its eigenvalue \lambda\in R, and is denoted as . It is an n\times n matrix of zeroes everywhere except for the diagonal, which is filled with \lambda and for the superdiagonal, which is composed of ones. Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix. This square matrix, consisting of diagonal blocks, can be compactly indicated as J_\oplus \cdots \oplus J_ or \mathrm\le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Polynomial (linear Algebra)
Minimal may refer to: * Minimal (music genre), art music that employs limited or minimal musical materials * "Minimal" (song), 2006 song by Pet Shop Boys * Minimal (supermarket) or miniMAL, a former supermarket chain in Germany and Poland * Minimal (''Dungeons & Dragons''), a creature of magically reduced size in the game ''Dungeons & Dragons'' * Minimal (chocolate), a bean to bar chocolate store in Japan, featured in '' Kantaro: The Sweet Tooth Salaryman'' * Minimal (clothing), an Indonesia clothing-retail company that worked with fashion model Ayu Gani * MINIMAL (restaurant), high end restaurant in Taichung Taichung (, Wade–Giles: '), officially Taichung City, is a special municipality (Taiwan), special municipality in central Taiwan. Taichung is Taiwan's second-largest city, with more than 2.85 million residents, making it the largest city in Ce ..., Taiwan See also * * Minimalism (other) * Maximal (other) * Minimisation (other) * Minimal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Radius
''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Mark Clyne, with Max Martini, Emily Mortimer, Clayne Crawford, and Bruce Greenwood in supporting roles. The film is set in a civil war-ridden Moldova as invisible entities slaughter any living being caught in their path. The film was released worldwide on December 9, 2016 on Netflix. On February 1, 2017, Netflix released a prequel graphic novel of the film called ''Spectral: Ghosts of War'' which was made available digitally through the website ComiXology. Plot DARPA researcher Mark Clyne is sent to a United States, US United States Armed Forces, military Air base, airbase on the outskirts of Chișinău, to consult his created line of hyperspectral imaging goggles issued to United States Army, US Army United States Army Special Forces, S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evident (f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said to converge absolutely if \textstyle\sum_^\infty \left, a_n\ = L for some real number \textstyle L. Similarly, an improper integral of a function, \textstyle\int_0^\infty f(x)\,dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if \textstyle\int_0^\infty , f(x), dx = L. A convergent series that is not absolutely convergent is called conditionally convergent. Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally convergen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form \sum_^\infty a_nx^n=a_0+a_1x+ a_2x^2+\cdots, where the a_n, called ''coefficients'', are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of Holomorphy
In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain. Formally, an open set \Omega in the ''n''-dimensional complex space ^n is called a ''domain of holomorphy'' if there do not exist non-empty open sets U \subset \Omega and V \subset ^n where V is connected, V \not\subset \Omega and U \subset \Omega \cap V such that for every holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ... f on \Omega there exists a holomorphic function g on V with f = g on U In the n=1 case, every open set is a domain of holomorphy: we can define a holomorphic function that is not identically zero, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is '' analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term '' analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Change Of Basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector on one basis is, in general, different from the coordinate vector that represents on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis. Such a conversion results from the ''change-of-basis formula'' which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Using matrices, this formula can be written :\mathbf x_\mathrm = A \,\mathbf x_\mathrm, where "old" and "new" refer respectively to the initially defined basis and the other basis, \mathbf x_\mathrm and \mathbf x_\mathrm are the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Eigenvector
In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let V be an n-dimensional vector space and let A be the matrix representation of a linear map from V to V with respect to some ordered basis. There may not always exist a full set of n linearly independent eigenvectors of A that form a complete basis for V. That is, the matrix A may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue \lambda_i is greater than its geometric multiplicity (the nullity of the matrix (A-\lambda_i I), or the dimension of its nullspace). In this case, \lambda_i is called a defective eigenvalue and A is called a defective matrix. A generalized eigenvector x_i corresponding to \lambda_i, together with the matrix (A-\lambda_i I) generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum Of Vector Spaces
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules. Construction for vector spaces and abelian groups We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
