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Golden–Thompson Inequality
In physics and mathematics, the Golden–Thompson inequality is a Trace inequalities, trace inequality between Matrix_exponential, exponentials of symmetric and Hermitian matrices proved independently by and . It has been developed in the context of statistical mechanics, where it has come to have a particular significance. Statement The Golden–Thompson inequality states that for (real) symmetric or (complex) Hermitian matrix, Hermitian matrices ''A'' and ''B'', the following trace inequality holds: : \operatorname\, e^ \le \operatorname \left(e^A e^B\right). This inequality is well defined, since the quantities on either side are real numbers. For the expression on right hand side of the inequality, this can be seen by rewriting it as \operatorname(e^e^B e^) using the Trace (linear algebra)#Cyclic property, cyclic property of the trace. Motivation The Golden–Thompson inequality can be viewed as a generalization of a stronger statement for real numbers. If ''a'' and ''b' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physic ... [...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\, _ = [\mathrm (, T, ^p). If T is compact and H_1,\,H_2 are separable, then : \, T\, _ := \bigg( \sum _ s^p_n(T)\bigg)^ for s_1(T) \ge s_2(T) \ge \cdots s_n(T) \ge \cdots \ge 0 the singular values of T, i.e. the eigenvalues of the Hermitian operator , T, :=\sqrt. Properties In the following we formally extend the range of p to ,\infty/math> with the convention that \, \cdot\, _ is the operator norm. The dual index to p=\infty is then q=1. * The Schatten norms are unitarily invariant: for unitary operators U and V and p\in ,\infty/math>, :: \, U T V\, _p = \, T\, _p. * They satisfy Hölder's inequality: for all p\in ,\infty/math> and q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Mathematical Physics
The ''Journal of Mathematical Physics'' is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in January 1960; it became a monthly publication in 1963. The current editor is Jan Philip Solovej from University of Copenhagen The University of Copenhagen ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in .... Its 2018 Impact Factor is 1.355 Abstracting and indexing This journal is indexed by the following services: 2013. References External links [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kostant Convexity Theorem
In mathematics, Kostant's convexity theorem, introduced by , states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a generalization of a result of , and for hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all ''n'' by ''n'' complex self-adjoint matrices with given eigenvalues Λ = (λ1, ..., λ''n'') is the convex polytope with vertices all permutations of the coordinates of Λ. Kostant used this to generalize the Golden–Thompson inequality to all compact groups. Compact Lie groups Let ''K'' be a connected compact Lie group with maximal torus ''T'' and Weyl group ''W'' = ''N''''K''(''T'')/''T''. Let their Lie algebras be \mathfrak and \mathfrak. Let ''P'' be the orthogonal projection of \mathfrak onto \mathfrak for some Ad-invariant inner product on \mathfrak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commuting Matrices
In linear algebra, two matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other. Characterizations and properties * Commuting matrices preserve each other's eigenspaces. As a consequence, commuting matrices over an algebraically closed field are simultaneously triangularizable; that is, there are bases over which they are both upper triangular. In other words, if A_1,\ldots,A_k commute, there exists a similarity matrix P such that P^ A_i P is upper triangular for all i \in \. The converse is not necessarily true, as the following counterexample shows: *:\begin 1 & 2 \\ 0 & 3 \end\begin 1 & 1 \\ 0 & 1 \end = \begin 1 & 3 \\ 0 & 3 \end \ne \begin 1 & 5 \\ 0 & 3 \end=\begin 1 & 1 \\ 0 & 1 \end\begin 1 & 2 \\ 0 & 3 \end. : However, if the square of the commutator of two matrices is zero, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commuting Matrices
In linear algebra, two matrices A and B are said to commute if AB=BA, or equivalently if their commutator ,B AB-BA is zero. A set of matrices A_1, \ldots, A_k is said to commute if they commute pairwise, meaning that every pair of matrices in the set commute with each other. Characterizations and properties * Commuting matrices preserve each other's eigenspaces. As a consequence, commuting matrices over an algebraically closed field are simultaneously triangularizable; that is, there are bases over which they are both upper triangular. In other words, if A_1,\ldots,A_k commute, there exists a similarity matrix P such that P^ A_i P is upper triangular for all i \in \. The converse is not necessarily true, as the following counterexample shows: *:\begin 1 & 2 \\ 0 & 3 \end\begin 1 & 1 \\ 0 & 1 \end = \begin 1 & 3 \\ 0 & 3 \end \ne \begin 1 & 5 \\ 0 & 3 \end=\begin 1 & 1 \\ 0 & 1 \end\begin 1 & 2 \\ 0 & 3 \end. : However, if the square of the commutator of two matrices is zero, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
