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Eigenvalue algorithm
An eigenvalue algorithm is a numerical method for computing eigenvalues of a matrix, using dense methods like the QR algorithm or iterative methods for large sparse matrices.
Eigenvalue interlacing
Eigenvalue interlacing is a linear algebra theorem stating that the eigenvalues of a Hermitian matrix and those of any principal submatrix must alternate in a fixed order.
Eigenvalues and eigenvectors
Eigenvalues and eigenvectors, also called characteristic roots, are scalars and nonzero vectors of a square matrix or operator satisfying Ax = λx, so the matrix only stretches the vector's length.
Einstein notation
Einstein notation, also called the Einstein summation convention, is a notational convention in which an index appearing twice in a term is implicitly summed, omitting the summation sign.
Eisenstein series
Eisenstein series are modular forms defined as sums of inverse powers over the nonzero points of a lattice, the simplest explicit examples in the theory of modular forms.
Eisenstein's criterion
Eisenstein's criterion, also called the Schönemann–Eisenstein theorem, is a test in algebra that proves a polynomial with integer coefficients irreducible over the rationals, published by Eisenstein in 1850.
Eldon Hansen
Eldon Hansen, full name Eldon Robert Hansen, is an American mathematician born in 1927, coauthor of Global Optimization Using Interval Analysis and namesake of interval methods.
Element (category theory)
In category theory, a generalized element of an object X is a morphism x : A → X, whose domain A is called its shape or stage of definition.
Elementary algebra
Elementary algebra, also known as college algebra or high school algebra, is the branch of mathematics that uses letters as variables to write and solve equations, taught in school and introductory college courses.
Elementary matrix
An elementary matrix, or elementary matrices collectively, is a square matrix made from the identity by one row operation: a swap, a scaling, or a row addition.
Elementary symmetric polynomial
An elementary symmetric polynomial σₖ is the sum of all products of k distinct variables x₁, ..., xₙ; by the fundamental theorem, every symmetric polynomial is uniquely a polynomial in them, a fact underlying Vieta's formulas.
Elimination theory
Elimination theory is the branch of algebra that removes unknowns from systems of multivariate polynomial equations, using resultants and Gröbner bases, with applications in equation solving, geometric theorem proving, and computer-aided design.
Elliptic curve
An elliptic curve is a smooth curve of genus one with a group law, given by y² = x³ + ax + b, used in modern cryptography.
Emmy Noether
Emmy Noether, born Amalie Emmy Noether, was a German mathematician whose 1918 theorems linked symmetries to conservation laws, and whose 1921 ideal theory founded modern abstract algebra.
Endomorphism
An endomorphism is a structure-preserving map from a mathematical object to itself, defined for groups, modules, rings, and vector spaces; in linear algebra it is a linear map from a space to itself.
Endomorphism ring
The endomorphism ring, also called the endomorphism algebra, is the ring formed by all homomorphisms from a mathematical object to itself, with composition of maps as multiplication.
Enriched category
An enriched category is a category-like structure whose hom-sets are replaced by objects of a monoidal base category, yielding ordinary categories, metric spaces, and 2-categories as instances.
Epsilon number
An epsilon number is a transfinite ordinal that is a fixed point of omega exponentiation, satisfying ω^α = α, with ε₀ the first and best known.
Equals sign
The equals sign, also called the equality sign, is a mathematical symbol stating that two expressions have the same value, introduced by Robert Recorde in 1557.
Equation
An equation is a mathematical statement that two expressions are equal, written with an equals sign; identities hold for all values, conditional equations for some, and contradictions for none.
Equation solving
Equation solving, also called solving an equation, is the task of finding values of unknowns that make equations true, either exactly in symbols or approximately by numerical methods.
Equiconsistency
Equiconsistency is a relation between formal theories in which each is consistent if and only if the other is; it underlies the consistency strength hierarchy comparing theories like ZFC.
Equivalence of categories
Equivalence of categories is the standard category-theoretic notion of sameness between two categories: a pair of functors going in opposite directions, inverse up to natural isomorphism.
Equivalence relation
An equivalence relation is a binary relation that is reflexive, symmetric, and transitive, so it behaves like equality; its equivalence classes form a partition of the set.
Erdős cardinal
An Erdős cardinal is a large cardinal defined by a partition property, named for Paul Erdős; the most-used case is κ(ω₁), and a Ramsey cardinal is the κ-Erdős cardinal.
Erdős–Bacon number
An Erdős–Bacon number is the sum of a person's Erdős number and Bacon number, their collaboration distances from mathematician Paul Erdős and actor Kevin Bacon through coauthored papers and shared film appearances.
Erdős–Kac theorem
The Erdős–Kac theorem, proved by Paul Erdős and Mark Kac in 1940, states that the number of distinct prime factors of an integer is normally distributed.
Erdős–Straus conjecture
The Erdős–Straus conjecture is an unsolved problem in number theory, posed by Paul Erdős and Ernst Straus in 1948, stating that 4/n is always a sum of three unit fractions.
Erdős–Tetali theorem
The Erdős–Tetali theorem is a 1990 result in additive number theory, proved by Paul Erdős and Prasad Tetali, that economical asymptotic bases of order k exist for every fixed k.
Erdős–Turán conjecture on additive bases
The Erdős–Turán conjecture on additive bases, posed by Paul Erdős and Pál Turán in 1941, is an open problem stating that representations cannot be bounded.