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Moore-Penrose Pseudoinverse
In mathematics, and in particular linear algebra, a pseudoinverse A+ of a matrix A is a generalization of the inverse matrix.[1] The most widely known type of matrix pseudoinverse is the Moore–Penrose inverse,[2][3][4][5] which was independently described by E. H. Moore[6] in 1920, Arne Bjerhammar[7] in 1951, and Roger Penrose[8] in 1955. Earlier, Erik Ivar Fredholm
Erik Ivar Fredholm
had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse. A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution (see below under § Applications)
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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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Linear Independence
In the theory of vector spaces, a set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent. These concepts are central to the definition of dimension.[1] A vector space can be of finite-dimension or infinite-dimension depending on the number of linearly independent basis vectors
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Zero Matrix
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero.[1] Some examples of zero matrices are 0 1 , 1 = [ 0 ] ,   0 2 , 2 = [ 0 0 0 0 ] ,   0 2 , 3 = [ 0 0 0 0 0 0 ] .   displaystyle 0_ 1,1 = begin bmatrix 0end bmatrix , 0_ 2,2 = begin bmatrix 0&0\0&0end bmatrix , 0_ 2,3 = begin bmatrix 0&0&0\0&0&0end bmatrix
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Projection (linear Algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged.[1] Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection
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Orthogonal Projector
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged.[1] Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection
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Range (mathematics)
In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage. Modern usage almost always uses range to mean image. The codomain of a function is some arbitrary super-set of image. In real analysis, it is the real numbers. In complex analysis, it is the complex numbers. The image of a function is the set of all outputs of the function
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Orthogonal Complement
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement
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Direct Sum Of Modules
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)
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Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain. Evaluating a function at each element of a subset X of the domain, produces a set called the image of X under or through the function. The inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S. Image and inverse image may also be defined for general binary relations, not just functions.Contents1 Definition1.1 Image of an element 1.2 Image of a subset 1.3 Image of a function2 Inverse image 3 Notation for image and inverse image3.1 Arrow notation 3.2 Star notation 3.3 Other terminology4 Examples 5 Consequences 6 See also 7 Notes 8 ReferencesDefinition[edit] The word "image" is used in three related ways
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Tikhonov Regularization
Tikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of ill-posed problems. In statistics, the method is known as ridge regression, in machine learning it is known as weight decay, and with multiple independent discoveries, it is also variously known as the Tikhonov–Miller method, the Phillips–Twomey method, the constrained linear inversion method, and the method of linear regularization
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Continuous Function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article. As an example, consider the function h(t), which describes the height of a growing flower at time t
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Matrix Norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).Contents1 Definition 2 Matrix norms induced by vector norms 3 "Entrywise" matrix norms3.1 L2,1 and Lp,q norms 3.2 Frobenius norm 3.3 Max norm4 Schatten norms 5 Consistent norms 6 Compatible norms 7 Equivalence of norms7.1 Examples of norm equivalence8 Notes 9 ReferencesDefinition[edit] In what follows, K displaystyle K will denote a field of either real or complex numbers. Let K m × n displaystyle K^ mtimes n denote the vector space of all matrices of size m × n displaystyle mtimes n (with m displaystyle m rows and n displaystyle n columns) with entries in the field
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Circulant Matrix
In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector
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Proofs Involving The Moore–Penrose Inverse
In linear algebra, the Moore-Penrose inverse is a matrix that satisfies some but not necessarily all of the properties of an inverse matrix
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Fourier Transform
The Fourier transform
Fourier transform
(FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes. The Fourier transform
Fourier transform
of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform
Fourier transform
is called the frequency domain representation of the original signal. The term Fourier transform
Fourier transform
refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time
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