* In the spectral decomposition of matrices, lambda indicates the diagonal matrix of the eigenvalues of the matrix ( mathematics ).

* Cylindrical algebraic decomposition, in mathematics

In mathematics, factorization ( also factorisation in British English ) or factoring is the decomposition of an object ( for example, a number, a polynomial, or a matrix ) into a product of other objects, or factors, which when multiplied together give the original.

In mathematics, a Voronoi diagram is a special kind of decomposition of a metric space, determined by distances to a specified family of objects ( subsets ) in the space.

In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:

In mathematics, Jordan decomposition may refer to

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational ( curl-free ) vector field and a solenoidal ( divergence-free ) vector field ; this is known as the Helmholtz decomposition.

In mathematics, a handle decomposition of an m-manifold M is a union

In mathematics, the Bruhat decomposition ( named after François Bruhat ) G = BWB into cells can be regarded as a general expression of the principle of Gauss – Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices — but with exceptional cases.

In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.

* Prime decomposition of integers, see fundamental theorem of arithmetic ( for the mathematics ) or integer factorization ( for applications )

In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number z as

In mathematics, the term cycle decomposition can mean:

Multipole moments in mathematics and mathematical physics form an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other.

* Levi decomposition in mathematics ( including Levi theorem, Levi subgroup, Levi subalgebra )

It should be stressed that the kinematical decomposition we are about to describe is pure mathematics valid for any Lorentzian manifold.

In mathematics, composition operators commonly occur in the study of shift operators, for example, in the Beurling-Lax theorem and the Wold decomposition.

In mathematics, Shannon's expansion or the Shannon decomposition is a method by which a Boolean function can be represented by the sum of two sub-functions of the original.

* Top ( mathematics ), in module theory, the largest semisimple quotient of a module

In mathematics, the term semisimple ( sometimes completely reducible ) is used in a number of related ways, within different subjects.

In mathematics, especially in the area of algebra known as ring theory, a semiprimitive ring is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring.

In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial F on V such that

In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group.

In mathematics, a semi-local ring is a ring for which R / J ( R ) is a semisimple ring, where J ( R ) is the Jacobson radical of R.

In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts.

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i. e., non-abelian Lie algebras whose only ideals are

In mathematics, in the field of representation theory, the Borel – Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups and complex semisimple Lie groups.

In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups.

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

Finite fields have applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type.

In mathematics, a Lie algebra (, not ) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

In mathematics, a Lie group () is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics.

Lie groups occur in abundance throughout mathematics and physics.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group ; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi ; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled

The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics.

Despite this they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups.

The theory of Lie groups describes continuous symmetry in mathematics ; its importance there and in theoretical physics ( for example quark theory ) grew steadily in the twentieth century.

The influence on graduate education in pure mathematics is perhaps most noticeable in the treatment now current of Lie groups and Lie algebras.

A key theme from the " categorical " point of view is that mathematics requires not only certain kinds of objects ( Lie groups, Banach spaces, etc.

From this time, and certainly much helped by Weyl's expositions, Lie groups and Lie algebras became a mainstream part both of pure mathematics and theoretical physics.

Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal group ), differential geometry ( Riemannian metric ), differential topology ( intersection forms of four-manifolds ), and Lie theory ( the Killing form ).

Since Lie groups ( and some analogues such as algebraic groups ) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied.

Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory ( such as singularity theory ).

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry.

One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real reductive Lie groups.

In mathematics, G < sub > 2 </ sub > is the name of three simple Lie groups ( a complex form, a compact real form and a split real form ), their Lie algebras, as well as some algebraic groups.