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Stahl's Theorem
In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions. It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture. In 2015, Alexandre Eremenko gave a simplified proof of Stahl's theorem. In 2023, Otte Heinävaara proved a structure theorem for Hermitian matrices introducing tracial joint spectral measures that implies Stahl's theorem as a corollary. Statement of the theorem Let \operatorname denote the trace of a matrix. If A and B are n\times n Hermitian matrices and B is positive semidefinite, define \mathbf(t) = \operatorname(\exp(A-tB)), for all real t\geq 0. Then \mathbf can be represented as the Laplace transform of a non-negative Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Analysis
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory). Matrix spaces The set of all ''m'' × ''n'' matrices over a field ''F'' denoted in this article ''M''''mn''(''F'') form a vector space. Examples of ''F'' include the set of rational numbers \mathbb, the real numbers \mathbb, and set of complex numbers \mathbb. The spaces ''M''''mn''(''F'') and ''M''''pq''(''F'') are different spaces if ''m'' and ''p'' are unequal, and if ''n'' and ''q'' are unequal; for instance ''M''32(''F'') ≠ ''M''23(''F''). Two ''m''&thins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Robert Stahl
Herbert may refer to: People Individuals * Herbert (musician), a pseudonym of Matthew Herbert Name * Herbert (given name) * Herbert (surname) Places Antarctica * Herbert Mountains, Coats Land * Herbert Sound, Graham Land Australia * Herbert, Northern Territory, a rural locality * Herbert, South Australia. former government town * Division of Herbert, an electoral district in Queensland * Herbert River, a river in Queensland * County of Herbert, a cadastral unit in South Australia Canada * Herbert, Saskatchewan, Canada, a town * Herbert Road, St. Albert, Canada New Zealand * Herbert, New Zealand, a town * Mount Herbert (New Zealand) United States * Herbert, Illinois, an unincorporated community * Herbert, Michigan, a former settlement * Herbert Creek, a stream in South Dakota * Herbert Island, Alaska Arts, entertainment, and media Fictional entities * Herbert (Disney character) * Herbert Pocket (''Great Expectations'' character), Pip's close friend and roommate in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Seiringer
Robert Seiringer (1 September 1976, Vöcklabruck) is an Austrian mathematical physicist. Life and work Seiringer studied physics at the University of Vienna, where in 1999 he acquired his diploma and in 2000 with Jakob Yngvason as thesis advisor attained a doctorate. In 2005 he attained his habilitation qualification at the University of Vienna. With a Schrödinger scholarship, he went in 2001 to Princeton University. There he became in 2003 assistant professor. Starting from 2010 he is an associate professor at McGill University. In addition he is extraordinarius professor at the University of Vienna. Seiringer made substantial progress in the mathematical theory of quantum gases and particularly Bose–Einstein condensate (BEC). He partly proved the existence of BEC for interacting boson gases in the Gross–Pitaevskii limit in collaboration with Elliott Lieb. They proved also superfluidity in this limit and derived the Gross–Pitaevskii equation in the special case of BEC ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandre Eremenko
Alexandre Eremenko (born 1954 in Kharkiv, Ukraine; ua, Олександр Емануїлович Єременко, transcription: Olexandr Emanuilowitsch Jeremenko) is a Ukrainian-American mathematician who works in the fields of complex analysis and dynamical systems. He is a grandnephew of a Marshal of the Soviet Union Andrey Yeryomenko. Academic career Eremenko was born into a medical family. His father Emmanuel Berger was a pathophysiologist, professor and head of the Department of pathophysiology at Ternopil National Medical University. His mother Neonila Eremenko was an ophthalmologist. He obtained his master's degree from Lviv University in 1976 and worked in the Institute of Low temperature physics and Engineering in Kharkiv until 1990. He received his PhD from Rostov State University in 1979 ''(Asymptotic Properties of Meromorphic and Subharmonic Functions),'' and is currently a distinguished professor at Purdue University. In complex dynamics, Eremenko explored es ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otte Heinävaara
Otte is a surname and given name. Notable persons with that name include: Given name *Otte Brahe (1518–1571), Danish (Scanian) nobleman and statesman *Otte Krumpen (1473–1569), Marshal of Denmark from 1554 to 1567 *Otte Rømer (c.1330–1409), Norwegian nobleman, state councilor, and landowner *Otte Rud (1520–1565), Danish admiral during the Northern Seven Years' War *Otte Wallish (1903–1977), Czech-Israeli graphic designer Surname * Carl Otte (1924–2011), American politician *Carlo Otte (1908–?), German Nazi administrator *Charles Otte (born 1956), American theatre director, producer, designer and educator *Christian Otte (1943 –2005), Belgian painter *Clifford Otte (born 1933), Wisconsin politician * Dan Otte (born 1939), American behavioral ecologist * Eileen Otte (born 1922), American model agency executive *Elise Otté (1818–1903), Anglo-Danish linguist, scholar and historian *Friedrich-Wilhelm Otte (1898–1944), German Wehrmacht general *Gary Otte (1971-2017), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrices
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although note that in quantum mechanics, A^\ast typically means the complex conjugate only, and not the conjugate transpose. Alternative characterizations Hermit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although note that in quantum mechanics, A^\ast typically means the complex conjugate only, and not the conjugate transpose. Alternative characterizations Hermit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Semidefinite 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] |
