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Frobenius's Theorem (group Theory)
There are several mathematical theorems named after Ferdinand Georg Frobenius. They include: * Frobenius theorem (differential topology) in differential geometry and topology for integrable subbundles * Frobenius theorem (real division algebras) in abstract algebra characterizing the finite-dimensional real division algebras * Frobenius reciprocity theorem in group representation theory describing the reciprocity relation between restricted and induced representations on a subgroup * Perron–Frobenius theorem In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive component ... in matrix theory concerning the eigenvalues and eigenvectors of a matrix with positive real coefficients * Frobenius's theorem (group theory) about the number of solutions of ''x''''n''=1 in a group {{disambig Mathematics d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants), and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds. Biography Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Theorem (differential Topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of ''r'' vector fields mesh into coordinate grids on ''r''-dimensional integral manifolds. The theorem is foundational in differential topology and calculus on manifolds. Introduction In its most elementary form, the theorem addresses the problem of finding a maximal set of inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Theorem (real Division Algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: * (the real numbers) * (the complex numbers) * (the quaternions). These algebras have real dimension , and , respectively. Of these three algebras, and are commutative, but is not. Proof The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra. Introducing some notation * Let be the division algebra in question. * Let be the dimension of . * We identify the real multiples of with . * When we write for an element of , we tacitly assume that is contained in . * We can consider as a finite-dimensional -vector space. Any element of defines an endomorphism of by left-multiplication, we identify with that endomorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perron–Frobenius Theorem
In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems ( subshifts of finite type); to economics ( Okishio's theorem, Hawkins–Simon condition); to demography ( Leslie population age distribution model); to social networks ( DeGroot learning process); to Internet search engines (PageRank); and even to ranking of football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. Statement Let positive and non-negative respectively describe matrices with exclusively positive real numbers as elements and matrices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius's Theorem (group Theory)
There are several mathematical theorems named after Ferdinand Georg Frobenius. They include: * Frobenius theorem (differential topology) in differential geometry and topology for integrable subbundles * Frobenius theorem (real division algebras) in abstract algebra characterizing the finite-dimensional real division algebras * Frobenius reciprocity theorem in group representation theory describing the reciprocity relation between restricted and induced representations on a subgroup * Perron–Frobenius theorem In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive component ... in matrix theory concerning the eigenvalues and eigenvectors of a matrix with positive real coefficients * Frobenius's theorem (group theory) about the number of solutions of ''x''''n''=1 in a group {{disambig Mathematics d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |