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Singular Value
In mathematics, in particular functional analysis, the singular values of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator T^*T (where T^* denotes the adjoint of T). The singular values are non-negative real numbers, usually listed in decreasing order (''σ''1(''T''), ''σ''2(''T''), …). The largest singular value ''σ''1(''T'') is equal to the operator norm of ''T'' (see Min-max theorem). If ''T'' acts on Euclidean space \Reals ^n, there is a simple geometric interpretation for the singular values: Consider the image by T of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of T (the figure provides an example in \Reals^2). The singular values are the absolute values of the eigenvalues of a normal matrix ''A'', because the spectral theorem can be applied to obtain unitary diagonalization of A as A = U\Lambd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a Matrix decomposition, factorization of a real number, real or complex number, complex matrix (mathematics), matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition#Matrix polar decomposition, polar decomposition. Specifically, the singular value decomposition of an m \times n complex matrix is a factorization of the form \mathbf = \mathbf, where is an complex unitary matrix, \mathbf \Sigma is an m \times n rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \mathbf V^* is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrix, orthogonal matrices; in such contexts, the SVD ... [...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] |
Schatten Norm
In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of ''p''-integrability similar to the trace class norm and the Hilbert–Schmidt norm. Definition Let H_1, H_2 be Hilbert spaces, and T a (linear) bounded operator from H_1 to H_2. For p\in T\, _p = T, ^p), where , T, :=\sqrt, using the Square_root_of_a_matrix#Square_roots_of_positive_operators, operator square root. If T is compact and H_1,\,H_2 are separable, then : \, T\, _p := \bigg( \sum_ s^p_n(T)\bigg)^ for s_1(T) \ge s_2(T) \ge \cdots \ge s_n(T) \ge \cdots \ge 0 the singular values of T, i.e. the eigenvalues of the Hermitian operator , T, :=\sqrt. Special cases *The Schatten 1-norm is the nuclear norm (also known as the trace norm, or the Ky Fan ''n''-norm). * The Schatten 2-norm is the Frobenius norm. * The Schatten ∞-norm is the spectral norm (also known as the operator norm, or the largest singular value). Properties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Krein
Mark Grigorievich Krein (, ; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory. He was born in Kyiv, leaving home at age 17 to go to Odesa. He had a difficult academic career, not completing his first degree and constantly being troubled by anti-Semitic discrimination. His supervisor was Nikolai Chebotaryov. He was awarded the Wolf Prize in Mathematics in 1982 (jointly with Hassler Whitney), but was not allowed to attend the ceremony. David Milman, Mark Naimark, Israel Gohberg, Vadym Adamyan, Mikhail Livsic and other known mathematicians were his students. He died in Odesa. On 14 January 2008, the memorial plaque of Mark Krein was unveiled on the main administration building of I.I. Mechnikov Odesa Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Gohberg
Israel Gohberg (; ; 23 August 1928 – 12 October 2009) was a Bessarabian-born Soviet and Israeli mathematician, most known for his work in operator theory and functional analysis, in particular linear operators and integral equations. Biography Gohberg was born in Tarutino to parents Tsudik and Haya Gohberg. His father owned a small typography shop and his mother was a midwife. The young Gohberg studied in a Hebrew school in Taurtyne and then a Romanian school in Orhei, where he was influenced by the tutelage of Modest Shumbarsky, a student of the renowned topologist Karol Borsuk. He studied at the Kyrgyz Pedagogical Institute in Bishkek and at Moldova State University in Chișinău, completed his doctorate at Leningrad State University on a thesis advised by Mark Krein (1954), and attended Moscow State University for his habilitation degree. Gohberg joined the faculty at the Teacher's college in Soroca and the Teachers college in Bălți before returning to Chișinău where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erhard Schmidt
Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (), in the Governorate of Livonia (now Estonia). Mathematics His advisor was David Hilbert and he was awarded his doctorate from University of Göttingen in 1905. His doctoral dissertation was entitled ' and was a work on integral equations. Together with David Hilbert he made important contributions to functional analysis. Ernst Zermelo credited conversations with Schmidt for the idea and method for his classic 1904 proof of the Well-ordering theorem from an "Axiom of choice", which has become an integral part of modern set theory. After the war, in 1948, Schmidt founded and became the first editor-in-chief of the journal '. National Socialism During World War II Schmidt held positions of authority at the University of Berlin and had to carry out various Nazi resoluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl's Inequality
In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix. Weyl's inequality about perturbation Let A,B be Hermitian on inner product space V with dimension n, with spectrum ordered in descending order \lambda_1 \geq ... \geq \lambda_n. Note that these eigenvalues can be ordered, because they are real (as eigenvalues of Hermitian matrices). Weyl's inequality states that the spectrum of Hermitian matrices is stable under perturbation. Specifically, we have: In jargon, it says that \lambda_k is Lipschitz-continuous on the space of Hermitian matrices with operator norm. Weyl's inequality between eigenvalues and singular values Let A \in \mathbb^ have singular values \sigma_1(A) \geq \cdots \geq \sigma_n(A) \geq 0 and eigenvalues ordered so that , \lambda_1(A), \geq \cdots \geq , \lambda_n(A), . Then : , \lambda_1(A) \cdots \lambd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Royal Johnson
Charles Royal Johnson (born January 28, 1948) is an American mathematician specializing in linear algebra. He was a Class of 1961 professor of mathematics at College of William and Mary. The books ''Matrix Analysis'' and ''Topics in Matrix Analysis'', co-written by him with Roger Horn, are standard texts in advanced linear algebra. Career Johnson received a B.A. with distinction in Mathematics and Economics from Northwestern University in 1969. In 1972, he received a Ph.D. in Mathematics and Economics from the California Institute of Technology, where he was advised by Olga Taussky Todd; his dissertation was entitled "Matrices whose Hermitian Part is Positive Definite". Johnson held various professorships over ten years at the University of Maryland, College Park starting in 1974. He was a professor at Clemson University from 1984 to 1987. He was a professor of mathematics at the College of William and Mary The College of William & Mary (abbreviated as W&M) is a public re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Horn
Roger Alan Horn (born January 19, 1942) is an American mathematician specializing in matrix analysis. He was research professor of mathematics at the University of Utah. He is known for formulating the Bateman–Horn conjecture with Paul T. Bateman on the density of prime number values generated by systems of polynomials. His books ''Matrix Analysis'' and ''Topics in Matrix Analysis'', co-written with Charles R. Johnson, are standard texts in advanced linear algebra. Career Roger Horn graduated from Cornell University with high honors in mathematics in 1963, after which he completed his PhD at Stanford University in 1967. Horn was the founder and chair of the Department of Mathematical Sciences at Johns Hopkins University from 1972 to 1979. As chair, he held a series of short courses for a monograph series published by the Johns Hopkins Press. He invited Gene Golub and Charles Van Loan to write a monograph, which later became the seminal ''Matrix Computations'' text book. He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Matrix
In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : :A \text \iff A^*A = AA^* . The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis. The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix satisfying the equation is diagonalizable. (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus A = U D U^* and A^* = U D^* U^*where D is a diagonal matrix whose diagonal values are in general complex. The left and right singular vectors in the singular value decomposition of a normal matrix A = U D V^* dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
