Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. In basis-free terms, for any set of mutually commuting semisimple operators on a finite-dimensional complex vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. To each such basis vector, there is a function assigning each element of S its corresponding eigenvalue. Such a function is called a weight. For the square matrix section, see square matrix. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if for all x and y in S. Otherwise, the operation * is noncommutative. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... This article is about operators in mathematics, for other kinds of operators see operator (disambiguation). ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

Examples

Suppose the elements of S form a topological group isomorphic to the real numbers under addition. A weight is then a continuous additive-to-multiplicative homomorphism φ: R→C^×. It is easy to see that all such homomorphisms are of the form φ = φ_y for some y in C, where In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. ...

φ_y(x) = e^2πixy.

More generally, if S is a real vector space W, any continuous homomorphism from S to C^× is given by a vector y in the complexification of the dual space W^* of W. The homomorphism φ_y will be unitary (i.e., have absolute value 1 for all x in W) if any only if y lies in W^* itself. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In government, see Unitary state In mathematics, see Unitary matrix Unitary operator Unitary group Unitary representation This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... The graph of the absolute value function In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. ...

This situation arises typically in the representation theory of Lie algebras. If S is an abelian subalgebra of a real Lie algebra g (i.e., the Lie bracket of any two elements of S is 0) and V is a representation space of g, we obtain a set of mutually commuting operators on V indexed by S. For example, S could be a Cartan subalgebra. If we choose S judiciously, we can arrange that these operators should be semi-simple. Therefore, V determines a set of weights (with multiplicities) in the (possibly complexified) dual space of S. In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Cartan subalgebra is a certain kind of subalgebra of a Lie algebra. ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ... In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

Alternatively, if S is the topological group S¹, i.e., a circle, which we identify with the unit circle in the complex plane, a weight on S is given by an integer m: φ_m(s) = s^m. More generally, if S is a compact connected commutative Lie group (and therefore isomorphic to the n-torus (S¹)ⁿ for some n), the possible weights of S are given by n-tuples of integers. This situation arises typically in the representation theory of compact Lie groups, where S is typically taken to be a maximal torus, i.e., a maximal compact connected commutative Lie group. Several specialized usages of the terms compact and compactness exist. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... // Geometry In geometry, a torus (pl. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups. ...