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In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...
This article gives an overview of the various ways to multiply matrices. ...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...
In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with norm 1 under multiplication. All the unitary groups contain copies of this group. In mathematics, the circle group is the group of all complex numbers on the unit circle under multiplication. ...
The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of complex n×n skew-Hermitian matrices, with the Lie bracket given by the commutator. In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
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 linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ...
A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if and only...
Properties
Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group homomorphism In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
The kernel of this homomorphism is the set of unitary matrices with unit determinant. This subgroup is called the special unitary group, denoted SU(n). We then have a short exact sequence of Lie groups: In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...
In mathematics, the special unitary group of degree is the group of by unitary matrices with determinant and entries from the field of complex numbers, with the group operation that of matrix multiplication. ...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
This short exact sequence splits so that U(n) may written as a semidirect product of SU(n) by U(1). Here the U(1) subgroup of U(n) is consists of matrices of the form . In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ...
The unitary group U(n) is nonabelian for n > 1. The center of U(n) is the set of scalar matrices λI with λ ∈ U(1). This follows from Schur's lemma. The center is then isomorphic to U(1). Since the center of U(n) is a 1-dimensional abelian normal subgroup of U(n), the unitary group is not semisimple. Please refer to group theory for a general description of the topic. ...
In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if...
In mathematics, Schurs lemma is now a generic term applied to theorems on the commutant of a module M that is simple. ...
In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...
Topology The unitary group U(n) is endowed with the relative topology as a subset of Mn(C), the set of all n×n complex matrices, which is itself homeomorphic to a 2n2-dimensional Euclidean space. In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology). ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
As a topological space, U(n) is both compact and connected. The compactness of U(n) follows from the Heine-Borel theorem and the fact that it is a closed and bounded subset of Mn(C). To show that U(n) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...
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 mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Émile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ...
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...
A path in U(n) from the identity to A is then given by In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X f : I → X. The initial point of the path is f(0) and the terminal point is f(1). ...
Although it is connected, the unitary group is not simply connected. The first unitary group U(1) is topologically a circle, which is well known to have a fundamental group isomorphic to Z. In fact, the fundamental group of U(n) is infinite cyclic for all n: A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
One can show that the determinant map det : U(n) → U(1) induces an isomorphism of fundamental groups.
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