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In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... This picture illustrates how the hours in a clock form a group. ... 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... This article gives an overview of the various ways to perform matrix multiplication. ... 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 is the set of n×n invertible matrices, together with the operation 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, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ...


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 is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a Lie algebra is an algebraic structure whose main use is 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. ... In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...


The general unitary group consists of all matrices A such that AA * is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.

Contents

Properties

Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group homomorphism In 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...

detcolon mbox{U}(n) to mbox{U}(1)

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 n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... 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. ...

1tombox{SU}(n)tombox{U}(n)tombox{U}(1)to 1

This short exact sequence splits so that U(n) may be written as a semidirect product of SU(n) by U(1). Here the U(1) subgroup of U(n) consists of matrices of the form mbox{diag}(e^{itheta},1,1,ldots,1). 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... 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). ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...


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 subset of Euclidean space Rn is called compact if it is closed and bounded. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... 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 = S,mbox{diag}(e^{itheta_1},dots,e^{itheta_n}),S^{-1}.

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). ...

tmapsto S,mbox{diag}(e^{ittheta_1},dots,e^{ittheta_n}),S^{-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. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

pi_1(U(n)) cong mathbb Z.

One can show that the determinant map det : U(n) → U(1) induces an isomorphism of fundamental groups.


Classifying space

The classifying space for U(n) is described in the article classifying space for U(n). In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial. ... In mathematics, the classifying space for U(n) may be constructed as the Grassmanian of n-planes in an infinite-dimensional complex Hilbert space. ...


See also


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