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Theorem in linear algebra
matrix theory, the Perron–Frobenius theorem, proved in its first part by Oskar Perron (1907) and extended by Georg Frobenius (1912), asserts that a real
Perron–Frobenius_theorem
Topics referred to by the same term
There are several mathematical theorems named after Ferdinand Georg Frobenius. They include: Frobenius theorem (differential topology) in differential
Frobenius_theorem
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Theorem in abstract algebra
mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative
Frobenius theorem (real division algebras)
Frobenius_theorem_(real_division_algebras)
Number of solutions of linear systems in terms of matrix ranks
Russia; Rouché–Fontené theorem in France; Rouché–Frobenius theorem in Spain and many countries in Latin America; Frobenius theorem in Czechia and Slovakia
Rouché–Capelli_theorem
German mathematician (1849–1917)
Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg, a suburb of Berlin, from parents Christian Ferdinand Frobenius, a Protestant parson
Ferdinand_Georg_Frobenius
Theorem of group theory
n {\displaystyle n} . It was introduced by Frobenius (1903). A more general version of Frobenius's theorem states that if C {\displaystyle C} is a conjugacy
Frobenius's theorem (group theory)
Frobenius's_theorem_(group_theory)
Operator encoding information about iterated map
or the Perron–Frobenius operator or Ruelle–Perron–Frobenius operator, in reference to the applicability of the Perron–Frobenius theorem to the determination
Transfer_operator
Generalization of the Perron–Frobenius theorem to Banach spaces
In functional analysis, the Krein–Rutman theorem is a generalisation of the Perron–Frobenius theorem to infinite-dimensional Banach spaces. It was proved
Krein–Rutman_theorem
its degree. Frobenius proved this surprising conjecture, and it became known as the Frobenius determinant theorem. His proof of the theorem sparked a new
Frobenius_determinant_theorem
Mathematical problem
problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that
Coin_problem
Four-dimensional number system
)\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).} According to the Frobenius theorem, the algebra H {\displaystyle \mathbb {H} } is one of only two finite-dimensional
Quaternion
Concept in mathematics
G. Frobenius. Suppose G is a Frobenius group consisting of permutations of a set X. A subgroup H of G fixing a point of X is called a Frobenius complement
Frobenius_group
Foundational result in symplectic geometry
Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It
Darboux's_theorem
Control theory for nonlinear or time-variant systems
stability criterion for linear systems) The Popov criterion. The Frobenius theorem is a deep result in differential geometry. When applied to nonlinear
Nonlinear_control
Formula for number of orbits of a group action
sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, or the orbit-counting theorem, is a result in group theory that is often
Burnside's_lemma
Property of certain dynamical systems
ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable
Integrable_system
Economic theorem regarding rate of profit
most easily understood as an application of the Perron–Frobenius theorem. This latter theorem comes from a branch of linear algebra known as the theory
Okishio's_theorem
Existence and uniqueness of solutions to initial value problems
for all time. Mathematics portal Cauchy–Kovalevskaya theorem Complete vector fields Frobenius theorem (differential topology) Integrability conditions for
Picard–Lindelöf_theorem
Describes statistically the splitting of primes in a given Galois extension of Q
splitting of primes. Georg Frobenius established the framework for investigating this pattern and proved a special case of the theorem. The general statement
Chebotarev_density_theorem
as Frobenius morphism, Frobenius map) Frobenius determinant theorem Frobenius formula Frobenius group Frobenius complement Frobenius kernel Frobenius inner
List of things named after Ferdinand Georg Frobenius
List_of_things_named_after_Ferdinand_Georg_Frobenius
Duality between the process of restricting and inducting in representation theory
In mathematics, and in particular representation theory, Frobenius reciprocity is a theorem expressing a duality between the process of restricting and
Frobenius_reciprocity
Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
{\displaystyle d} nonzero terms. Alternative proofs use Helly's theorem or the Perron–Frobenius theorem. For any nonempty P ⊂ R d {\displaystyle P\subset \mathbb
Carathéodory's theorem (convex hull)
Carathéodory's_theorem_(convex_hull)
Bound on eigenvalues
satisfied here. For matrices with non-negative entries, see Perron–Frobenius theorem. Doubly stochastic matrix Hurwitz-stable matrix – Matrix whose eigenvalues
Gershgorin_circle_theorem
A prime p divides a^p–a for any integer a
Fermat quotient Frobenius endomorphism p-derivation Fractions with prime denominators: numbers with behavior relating to Fermat's little theorem RSA Table of
Fermat's_little_theorem
theorem (geometric group theory) Focal subgroup theorem (abstract algebra) Frobenius determinant theorem (group theory) Frobenius reciprocity theorem
List_of_theorems
{\displaystyle X} is nonzero. The theorem is also known as straightening out of a vector field. The Frobenius theorem in differential geometry can be considered
Straightening theorem for vector fields
Straightening_theorem_for_vector_fields
other information. Frobenius theorem. This fundamental theorem was stated and proved in 1840 by Feodor Deahna. Even though Frobenius cited Deahna's paper
List_of_misnamed_theorems
Branch of geometry
foliation on the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of
Contact_geometry
Method for producing composition algebras
finite-dimensional normed division algebras over the real numbers, while the Frobenius theorem states that the first three are the only finite-dimensional associative
Cayley–Dickson_construction
Non-associative algebras with positive-definite quadratic form
must be diagonal. Multiplicative quadratic form Radon–Hurwitz number Frobenius Theorem See: Lam 2005 Rajwade 1993 Shapiro 2000 See: Eckmann 1989 Eckmann
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Algebraic structure with "nice" duality properties
duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama
Frobenius_algebra
Condition under which an odd prime is a sum of two squares
case the Frobenius endomorphism of Z[i]/(p) is the identity. Kummer had already established that if f ∈ {1,2} is the order of the Frobenius automorphism
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Mapping theorem in topology
{\displaystyle k} . The Frobenius endomorphism of X ¯ {\displaystyle {\bar {X}}} (often the geometric Frobenius, or just the Frobenius), denoted by F q {\displaystyle
Lefschetz_fixed-point_theorem
projectives and injectives coincide. It is an analog of a Frobenius algebra. The stable category of a Frobenius category is canonically a triangulated category
Frobenius_category
Random process independent of past history
vector with all entries equal to 1. This is stated by the Perron–Frobenius theorem. If, by whatever means, lim k → ∞ P k {\textstyle \lim _{k\to \infty
Markov_chain
Topics referred to by the same term
Sylvester's theorem or the Sylvester theorem may refer to any of several theorems named after James Joseph Sylvester: The Sylvester–Gallai theorem, on the
Sylvester's_theorem
Square matrix whose off-diagonal entries are nonnegative
because of the corresponding property for nonnegative matrices. Perron–Frobenius theorem Nonnegative matrices Delay differential equation M-matrix P-matrix
Metzler_matrix
Method for solving ordinary differential equations
In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order
Frobenius_method
Classification theorem in group theory
subgroups are of "Frobenius type", a slight generalization of Frobenius group, and in fact later on in the proof are shown to be Frobenius groups. They have
Feit–Thompson_theorem
Matrix whose eigenvalues have negative real part
real components, representing positive feedback. M-matrix Perron–Frobenius theorem, which shows that any Hurwitz matrix must have at least one negative
Hurwitz-stable_matrix
Theorem classifying finite simple groups
classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is
Classification of finite simple groups
Classification_of_finite_simple_groups
(differential) jet (mathematics) Contact (mathematics) jet bundle Frobenius theorem (differential topology) Integral curve Diffeomorphism Large diffeomorphism
List of differential geometry topics
List_of_differential_geometry_topics
Theorem on the orders of subgroups
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is
Lagrange's theorem (group theory)
Lagrange's_theorem_(group_theory)
Map raising elements to the pth power, in characteristic p
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with
Frobenius_endomorphism
Algebraic structure with addition, multiplication, and division
which multiplication is non-commutative). This result is known as the Frobenius theorem. The octonions O, for which multiplication is neither commutative
Field_(mathematics)
Hungarian and American mathematician and physicist (1903–1957)
satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue
John_von_Neumann
Theorems that help decompose a finite group based on prime factors of its order
specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow
Sylow_theorems
Square matrix used to represent a graph or network
above by the maximum degree. This can be seen as result of the Perron–Frobenius theorem, but it can be proved easily. Let v be one eigenvector associated
Adjacency_matrix
Concepts from linear algebra
transitions from one state to some other state of the system. The Perron–Frobenius theorem gives sufficient conditions for a Markov chain to have a unique dominant
Eigenvalues_and_eigenvectors
Algorithm used by Google Search to rank web pages
Normed eigenvectors exist and are unique by the Perron or Perron–Frobenius theorem. Example: consumers and products. The relation weight is the product
PageRank
Square matrices satisfy their characteristic equation
formal proof of the theorem in the general case of a matrix of any degree”. The general case was first proved by Ferdinand Frobenius in 1878. For a 1 ×
Cayley–Hamilton_theorem
Smooth manifold
important. For real-analytic J, the Newlander–Nirenberg theorem follows from the Frobenius theorem; for C∞ (and less smooth) J, analysis is required (with
Almost_complex_manifold
Theorem 23.16. James 2001, pp. 277, Corollary 23.17. G.Frobenius & I.Schur, Über die reellen Darstellungen der endlichen Gruppen (1906), Frobenius Gesammelte
Frobenius–Schur_indicator
Canonical form of matrices over a field
In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices
Frobenius_normal_form
Matrix in mathematics
identity matrix. For the non-singularity of A, according to the Perron–Frobenius theorem, it must be the case that s > ρ(B). Also, for a non-singular M-matrix
M-matrix
Rigidity theorem in differential geometry
uniquely determined up to a rigid motion of R3. Bonnet's theorem is a corollary of the Frobenius theorem, upon viewing the Gauss–Codazzi equations as a system
Bonnet_theorem
Theorem of dynamical systems
The Liouville–Arnold theorem is a result in classical mechanics which says, roughly speaking, that seemingly complicated systems can be described as combinations
Liouville–Arnold_theorem
Matrix used to describe the transitions of a Markov chain
also a stationary probability vector. On the other hand, the Perron–Frobenius theorem also ensures that every irreducible stochastic matrix has such a stationary
Stochastic_matrix
Mathematical space
non-degeneracy' condition called 'complete non-integrability'. From the Frobenius theorem, one recognizes the condition as the opposite of the condition that
3-manifold
Element of a unital algebra over the field of real numbers
the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition
Hypercomplex_number
Conjecture on zeros of the zeta function
function of a variety over a finite field correspond to eigenvalues of a Frobenius element on an étale cohomology group, the zeros of a Selberg zeta function
Riemann_hypothesis
Subset of a manifold that is a manifold itself; an injective immersion into a manifold
where immersed submanifolds provide the right context to prove the Frobenius theorem. An embedded submanifold (also called a regular submanifold) is an
Submanifold
Degree of connectedness within a graph
unique largest eigenvalue, which is real and positive, by the Perron–Frobenius theorem. This greatest eigenvalue results in the desired centrality measure
Centrality
for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal I {\displaystyle {\mathcal {I}}}
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
Theorem in algebraic geometry
simpler proofs of the Kempf vanishing theorem using the Frobenius morphism. Andersen, Henning Haahr (1980), "The Frobenius morphism on the cohomology of homogeneous
Kempf_vanishing_theorem
Measure in graph theory
the entries in the eigenvector be non-negative imply (by the Perron–Frobenius theorem) that only the greatest eigenvalue results in the desired centrality
Eigenvector_centrality
Topics referred to by the same term
curves Frobenius reciprocity theorem for group representations Stanley's reciprocity theorem for generating functions Reciprocity (engineering), theorems relating
Reciprocity_theorem
Mathematical theorem
{\displaystyle y''+p(x)y'+q(x)y=g(x)} has a solution expressible by a generalised Frobenius series when p ( x ) {\displaystyle p(x)} , q ( x ) {\displaystyle q(x)}
Fuchs's_theorem
Vector satisfying some of the criteria of an eigenvector
matrix Positive-(semi)definite Pfaffian Projection Spectral theorem Perron–Frobenius theorem Diagonal matrix, Triangular matrix, Tridiagonal matrix Block
Generalized_eigenvector
list of things named after Issai Schur. Frobenius–Schur indicator Herz–Schur multiplier Jordan–Schur theorem Lehmer–Schur algorithm Schur algebra Schur
List of things named after Issai Schur
List_of_things_named_after_Issai_Schur
Mathematical fallacy
freshman exponentiation, the child's binomial theorem, (rarely) the schoolboy binomial theorem, or the Frobenius identity is the generally-false equation (x + y)n = xn + yn
Freshman's_dream
Constrained least squares problem
problem above, and an active set method called TNT-NN. M-matrix Perron–Frobenius theorem Chen, Donghui; Plemmons, Robert J. (2009). Nonnegativity constraints
Non-negative_least_squares
normal subgroup of G, called the "Frobenius kernel", and G is the semidirect product of H and N (Frobenius' theorem). Lyndon, Roger C.; Schupp, Paul E
Malnormal_subgroup
In mathematics, a partition of a manifold into submanifolds
codimension n − 1 foliation). This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution
Foliation
Algebraic structure also called skew field
Wedderburn's little theorem: All finite division rings are commutative and therefore finite fields. (Ernst Witt gave a simple proof.) Frobenius theorem: The only
Division_ring
Result in mathematical economics on existence of a non-negative equilibrium output vector
Gantmacher as Kotelyanskiĭ lemma. Diagonally dominant matrix Perron–Frobenius theorem Sylvester's criterion Hawkins, David; Simon, Herbert A. (1949). "Some
Hawkins–Simon_condition
Distance function
metric and the Banach contraction principle to rederive the Perron–Frobenius theorem in finite-dimensional linear algebra and its analogues for integral
Hilbert_metric
Complex square matrix for which every principal minor is positive
Linear complementarity problem M-matrix Q-matrix Z-matrix Perron–Frobenius theorem Kellogg, R. B. (April 1972). "On complex eigenvalues ofM andP matrices"
P-matrix
Algebra over a field with only invertible elements and zero
numbers that are finite-dimensional as a vector space over R). The Frobenius theorem states that up to isomorphism there are three such algebras: the reals
Division_algebra
Describes the objects of a given type, up to some equivalence
multiplicities Frobenius normal form – Canonical form of matrices over a field (rational canonical form) Sylvester's law of inertia – Theorem of matrix algebra
Classification_theorem
Matrix with no negative elements
eigenvectors of square positive matrices are described by the Perron–Frobenius theorem. The trace and every row and column sum/product of a nonnegative matrix
Nonnegative_matrix
Mathematics, group theory
theorem had previously been proved by Burnside in 1897, Jordan in 1898, and Frobenius in 1902. John G. Thompson pointed out that a proof avoiding the use of
Burnside's_theorem
while Frobenius' theorem is stated in terms of convergent sequences and the epsilon-delta formulation of the limit of a function. Frobenius' theorem was
History_of_Grandi's_series
Metric used to rank web pages
{\displaystyle G^{*}} belong to the class of Perron–Frobenius operators and according to the Perron–Frobenius theorem the CheiRank P i ∗ {\displaystyle P_{i}^{*}}
CheiRank
Subbundle of the tangent bundle
automatically involutive. The converse is less trivial but holds by Frobenius theorem. Given any distribution Δ ⊆ T M {\displaystyle \Delta \subseteq TM}
Distribution (differential geometry)
Distribution_(differential_geometry)
Field theory theorem
primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies in particular
Primitive_element_theorem
positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal matrix, main diagonal Diagonalizable
Outline_of_linear_algebra
Topics referred to by the same term
southwestern France Ruelle operator Ruelle zeta function Ruelle-Perron-Frobenius theorem Ruel (disambiguation) This disambiguation page lists articles associated
Ruelle
Graph where each vertex has the same number of neighbors
multiplicity one. The "only if" direction is a consequence of the Perron–Frobenius theorem. There is also a criterion for regular and connected graphs : a graph
Regular_graph
which proved the theorem in special cases. To prove their result Schauenbug and Ng introduced the notion of 'generalied Frobenius–Schur' indicators,
Schauenburg–Ng_theorem
Stochastic matrix representing links between entities
CheiRank Arnoldi iteration Markov chain Transfer operator Perron–Frobenius theorem Web search engines Ermann, L.; Chepelianskii, A. D.; Shepelyansky
Google_matrix
Non-tensorial representation of the spin group
q}^{0}}(S)} is a finite-dimensional real division algebra. By the Frobenius theorem, it is therefore isomorphic to exactly one of R , C , H . {\displaystyle
Spinor
Random matrix with gaussian entries
theory), there is a 3-fold disjunction, which he traced back to the Frobenius theorem stating that there are only 3 real division algebras: the real, the
Gaussian_ensemble
Transformations induced by a mathematical group
known as the orbit–stabilizer theorem. If G is finite then the orbit–stabilizer theorem, together with Lagrange's theorem, gives | G ⋅ x | = [ G : G x
Group_action
More equations than unknowns (mathematics)
Least squares Moore–Penrose pseudoinverse Rouché-Capelli (or, Rouché-Frobenius) theorem Gentle, James E. (2012-12-06). Numerical Linear Algebra for Applications
Overdetermined_system
Type of algebraic number
Perron number. Perron numbers are named after Oskar Perron; the Perron–Frobenius theorem asserts that, for a real square matrix with positive algebraic entries
Perron_number
German mathematician
mathematician. He is known for providing proof of what is now known as Frobenius theorem in differential topology, which he published in Crelle's Journal in
Feodor_Deahna
Differential geometry technique
trivial group. The problem can now be handled by methods such as the Frobenius theorem. In other words, the algorithm has successfully terminated. On the
Cartan's_equivalence_method
algebraic integers all of whose conjugates (given by eigenvalues of the Frobenius endomorphism on the first cohomology group or Tate module) have absolute
Honda–Tate_theorem
FROBENIUS THEOREM
FROBENIUS THEOREM
FROBENIUS THEOREM
FROBENIUS THEOREM
Girl/Female
Hindu
Name of a Raga
Male
Finnish
Finnish form of German Hermann, HERMANNI means "army man."Â
Girl/Female
Christian & English(British/American/Australian)
First Rose
Boy/Male
Indian
Aim, Goal, End
Boy/Male
Gujarati, Hindu, Indian
Son of Yamraj (Lord of Death)
Boy/Male
Tamil
Nature
Female
Babylonian
, early ancestor of the gods.
Girl/Female
Tamil
Sushmita | ஸà¯à®·à¯à®®à®¿à®¤à®¾
Beautiful smile, Good smile
Boy/Male
Hindu
Hostage
Girl/Female
Arabic, Indian, Muslim, Traditional
Spirit; Angel; Candles
FROBENIUS THEOREM
FROBENIUS THEOREM
FROBENIUS THEOREM
FROBENIUS THEOREM
FROBENIUS THEOREM
a.
Containing many names or terms; multinominal; as, the polynomial theorem.
a.
Alt. of Theorematical
n.
The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.
n.
That which is considered and established as a principle; hence, sometimes, a rule.
n.
One who constructs theorems.
n.
A theorem or proposition so easy of demonstration as to be almost self-evident.
v. t.
To formulate into a theorem.
n.
A statement of a principle to be demonstrated.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
n.
A numerical coefficient in any particular case of the binomial theorem.
a.
Theorematic.