Classical automorphism groups

AbbreviationGeneral linear group (Classical automorphism groups)#

The general linear group \(\mathrm{GL}_n(R)\) is the group of invertible \(n \times n\) matrices with entries in a ring \(R\), under matrix multiplication. Over a field \(K\) it is the group of all invertible linear transformations of \(K^n\) written in coordinates, so once a basis is chosen, every group acting linearly on an \(n\)-dimensional vector space over \(K\) maps homomorphically into \(\mathrm{GL}_n(K)\). It is the basic example of a classical group: the special linear, orthogonal and unitary groups are all subgroups of it, and over finite fields it supplies a large family of finite groups.

Definition

Let \(R\) be a ring and \(n \ge 0\). The general linear group \(\mathrm{GL}_n(R)\) consists of the \(n \times n\) matrices \(A\) over \(R\) that have a two-sided inverse, a matrix \(B\) with \(AB = BA = I\), where \(I\) is the identity matrix. Products and inverses of invertible matrices are invertible, and matrix multiplication is associative, so these matrices form a group with identity \(I\). Equivalently, \(\mathrm{GL}_n(R)\) is the group of units of the matrix ring of \(n \times n\) matrices over \(R\).

When \(R\) is commutative, the determinant gives a practical test: a square matrix is invertible exactly when its determinant is a unit of \(R\). Over a field this means \(\det A \neq 0\); over the integers it means \(\det A = \pm 1\).

Examples
  • For \(n = 1\), a \(1 \times 1\) matrix is invertible exactly when its entry is a unit, so \(\mathrm{GL}_1(R)\) is the group of units \(R^\times\).
  • For \(n \ge 2\) and any ring with \(1 \neq 0\), the group is not commutative. The matrices $\(A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)$ are invertible, but \(AB\) has first row \((1, 1)\) while \(BA\) has first row \((0, 1)\). For larger \(n\), place \(A\) and \(B\) in the top-left corner of block-diagonal matrices whose remaining diagonal entries are \(1\); the products still differ.
  • Over the two-element field the group \(\mathrm{GL}_2(\mathbb{F}_2)\) has six elements and is not commutative, so it is isomorphic to the symmetric group on three letters.
  • \(\mathrm{GL}_n(\mathbb{Z})\) is the group of integer matrices with determinant \(\pm 1\); it is the group of changes of basis of the lattice \(\mathbb{Z}^n\).
Properties

Linear automorphisms. For a commutative ring \(R\), an invertible matrix acts on column vectors by multiplication, and this identifies \(\mathrm{GL}_n(R)\) with the group of invertible linear maps of \(R^n\) to itself. More generally, a choice of basis of a module \(V\) with \(n\) elements identifies \(\mathrm{GL}_n(R)\) with the group of invertible linear maps of \(V\); a different basis changes this identification by conjugation.

The determinant. For a commutative ring \(R\), the determinant is a group homomorphism from \(\mathrm{GL}_n(R)\) to \(R^\times\), and for \(n \ge 1\) it is surjective: the diagonal matrix with entries \(u, 1, \dots, 1\) has determinant \(u\). Its kernel is the special linear group: the matrices of determinant \(1\) are exactly the kernel of the determinant. Hence \(\mathrm{SL}_n(R)\) is a normal subgroup, and by the first isomorphism theorem \(\mathrm{GL}_n(R) / \mathrm{SL}_n(R) \cong R^\times\) when \(n \ge 1\).

The center. For a commutative ring \(R\), the center of \(\mathrm{GL}_n(R)\) consists of the scalar matrices \(uI\) with \(u\) a unit of \(R\): a matrix that commutes with every invertible matrix must be a multiple of the identity.

Change of ring. A homomorphism \(f \colon R \to S\) of commutative rings, applied entry by entry, gives a group homomorphism from \(\mathrm{GL}_n(R)\) to \(\mathrm{GL}_n(S)\). Reduction modulo a prime \(p\), from \(\mathrm{GL}_n(\mathbb{Z})\) to \(\mathrm{GL}_n(\mathbb{F}_p)\), is an instance.

Order over a finite field. If \(\mathbb{F}\) is a finite field with \(q\) elements, then

\[|\mathrm{GL}_n(\mathbb{F})| = (q^n - 1)(q^n - q)(q^n - q^2) \cdots (q^n - q^{n-1}),\]

the product of \(q^n - q^i\) over \(0 \le i < n\). A square matrix over a field is invertible exactly when its columns are linearly independent, so \(|\mathrm{GL}_n(\mathbb{F})|\) counts the ordered bases of \(\mathbb{F}^n\): the first column can be any nonzero vector (\(q^n - 1\) choices), and the \((i+1)\)-th column any vector outside the span of the first \(i\) (\(q^n - q^i\) choices). Since the determinant maps \(\mathrm{GL}_n(\mathbb{F})\) onto \(\mathbb{F}^\times\) for \(n \ge 1\), the first isomorphism theorem makes the index of \(\mathrm{SL}_n(\mathbb{F})\) equal to \(|\mathbb{F}^\times| = q - 1\), and Lagrange's theorem in the form \(|G| = [G : H]\,|H|\) gives \(|\mathrm{SL}_n(\mathbb{F})| \cdot |\mathbb{F}^\times| = |\mathrm{GL}_n(\mathbb{F})|\), so \(|\mathrm{SL}_n(\mathbb{F})| = |\mathrm{GL}_n(\mathbb{F})| / (q - 1)\).

The general linear group of a module is the coordinate-free version: the invertible linear maps of a module to itself, which are the same as its linear automorphisms. Its important subgroups include the special linear group, the orthogonal group and the unitary group. Over a linearly ordered commutative ring such as the real numbers, the matrices of positive determinant form a subgroup; over the reals it has index \(2\) when \(n \ge 1\). The real and complex general linear groups are Lie groups (see general linear group as a Lie group), and once a basis is chosen, a representation of a group on an \(n\)-dimensional vector space over \(K\) is a homomorphism from the group into \(\mathrm{GL}_n(K)\).

In Mathlib

Mathlib defines the general linear group as the group of units of the monoid of square matrices: an element is a matrix bundled together with a two-sided inverse. The rows and columns are labelled by an arbitrary finite type rather than by \(1, \dots, n\), so the usual \(\mathrm{GL}_n(R)\) is the case of an index set with \(n\) elements, and an empty index set gives the trivial group. The definition needs only a semiring, not a ring or a commutative ring. The description by determinants, the determinant homomorphism, the center and the identification with linear maps are stated for commutative rings, and the counting formula for finite fields. Cardinalities are natural numbers, with \(0\) for an infinite set, so the relation between the orders of the special and general linear groups holds for every commutative ring and every \(n \ge 1\): when \(\mathrm{GL}_n(R)\) is infinite, so is \(\mathrm{SL}_n(R)\) or \(R^\times\), and both sides are \(0\).

Written for Trough, under CC BY-SA 4.0.

Sources
MathlibMatrix.GeneralLinearGroup
Uses 8 · Used by 2
Statement usesGroup · Ring-valued matrix · Determinant · Ring · Invertible elements
Proof usesLagrange's theorem · First isomorphism theorem · Linear independence
Used byOrthogonal group · Unitary group

AbbreviationOrthogonal group#

The orthogonal group \(\mathrm{O}(n)\) is the group of \(n \times n\) matrices whose transpose is their inverse. Over the real numbers these are exactly the matrices of the linear maps of \(\mathbb{R}^n\) that preserve lengths and angles: the rotations, the reflections and their composites. It is one of the classical groups, it is the symmetry group of Euclidean geometry with a fixed origin, and it serves as a basic example of a compact Lie group.

Definition

Let \(R\) be a commutative ring. A square matrix \(A\) with entries in \(R\) is orthogonal if

\[A A^{\mathsf{T}} = I,\]

where \(A^{\mathsf{T}}\) is the transpose of \(A\) and \(I\) is the identity matrix. The orthogonal matrices of size \(n\) form the orthogonal group \(\mathrm{O}(n, R)\), a subgroup of the general linear group of invertible \(n \times n\) matrices; when \(R = \mathbb{R}\) one writes simply \(\mathrm{O}(n)\).

For square matrices over a commutative ring a one-sided inverse is automatically two-sided, so the condition can be written either way: \(A\) is orthogonal exactly when \(A A^{\mathsf{T}} = I\), and also exactly when \(A^{\mathsf{T}} A = I\). In terms of entries, the first condition says that the rows of \(A\) are orthonormal for the standard dot product, and the second says the same of the columns.

The set is a group under matrix multiplication: \(I\) is orthogonal, the inverse of \(A\) is \(A^{\mathsf{T}}\), which is again orthogonal, and if \(A\) and \(B\) are orthogonal then \((AB)(AB)^{\mathsf{T}} = A B B^{\mathsf{T}} A^{\mathsf{T}} = I\).

Examples
  • Size one. \(\mathrm{O}(1, R)\) consists of the elements \(a \in R\) with \(a^2 = 1\); over a field, or more generally an integral domain, this is \(\{1, -1\}\), a single element in characteristic \(2\).
  • Rotations of the plane. A \(2 \times 2\) matrix over a commutative ring is orthogonal with determinant \(1\) exactly when it has the form $\(\begin{pmatrix} a & -c \\ c & a \end{pmatrix}, \qquad a^2 + c^2 = 1.\)$ Over \(\mathbb{R}\) one can write \(a = \cos\theta\) and \(c = \sin\theta\), which gives the rotation by the angle \(\theta\). The remaining elements of \(\mathrm{O}(2)\) are the reflections in lines through the origin, with matrices of the form \(\begin{pmatrix} a & c \\ c & -a \end{pmatrix}\) where \(a^2 + c^2 = 1\).
  • Permutation matrices. Every permutation matrix is orthogonal, since its transpose is the matrix of the inverse permutation. More generally a matrix with exactly one nonzero entry in each row and column, that entry being \(\pm 1\), is orthogonal; over the integers these signed permutation matrices are the only orthogonal matrices.
  • Reflections. For a nonzero column vector \(v \in \mathbb{R}^n\), the matrix \(I - 2\,v v^{\mathsf{T}} / (v^{\mathsf{T}} v)\) is orthogonal. It is the reflection in the hyperplane orthogonal to \(v\).
Properties
Generalizations

Given a nondegenerate symmetric bilinear form on \(R^n\) with matrix \(B\), the matrices \(A\) satisfying \(A^{\mathsf{T}} B A = B\) form the orthogonal group of the form; the case \(B = I\) is the group described here. Indefinite forms on \(\mathbb{R}^n\) give the groups \(\mathrm{O}(p, q)\), among them the Lorentz group, and over a general field the orthogonal group of a quadratic form is studied in the same way. Replacing the transpose by the conjugate transpose gives the unitary group of complex matrices with \(A A^{*} = I\), of which the orthogonal group is the real counterpart.

In Mathlib

Mathlib defines the orthogonal group for an arbitrary commutative ring and for square matrices whose rows and columns are indexed by any finite type, not only by \(\{1, \dots, n\}\). It does not give a separate construction: the orthogonal group is the unitary group taken with respect to the trivial involution of the ring, so that the conjugate transpose becomes the plain transpose, and every general fact about unitary matrices applies to it. It is packaged as a set of matrices closed under multiplication and containing the identity, with the group structure, in which the inverse is the transpose, inherited from the unitary construction. The special orthogonal group is defined as the orthogonal matrices of determinant \(1\), and Mathlib notes that this is the intended group only when \(2\) is invertible in the ring. Coordinate-free statements, such as the generation by reflections, are phrased for linear isometries of an inner product space rather than for matrices.

Written for Trough, under CC BY-SA 4.0.

Sources
MathlibMatrix.orthogonalGroup
Uses 5 · Used by 1
Statement usesRing-valued matrix · Group · Subgroup · Ring · General linear group (Classical automorphism groups)
Used bySpecial orthogonal group

AbbreviationSpecial orthogonal group#

The special orthogonal group \(\mathrm{SO}_n(R)\) is the group of \(n \times n\) orthogonal matrices over a commutative ring \(R\) whose determinant equals \(1\). Over the real numbers it is the group of rotations of \(n\)-dimensional Euclidean space: the linear maps that preserve lengths, angles and orientation. It is one of the classical matrix groups, next to the general linear, special linear and unitary groups, and its small cases, the plane and spatial rotation groups \(\mathrm{SO}(2)\) and \(\mathrm{SO}(3)\), appear throughout geometry and mechanics.

Definition

Let \(R\) be a commutative ring and \(n \ge 0\). A square matrix \(A \in M_n(R)\) (an \(n \times n\) matrix over \(R\)) is orthogonal when its transpose is its inverse, that is, \(A^{\mathsf{T}} A = I\) and \(A A^{\mathsf{T}} = I\). For square matrices over a commutative ring one of the two equations implies the other: \(A\) is orthogonal exactly when \(A A^{\mathsf{T}} = I\), and exactly when \(A^{\mathsf{T}} A = I\). The special orthogonal group is

\[\mathrm{SO}_n(R) = \{\, A \in M_n(R) : A^{\mathsf{T}} A = I \text{ and } \det A = 1 \,\},\]

so a matrix lies in \(\mathrm{SO}_n(R)\) exactly when it is orthogonal and has determinant \(1\). Over \(\mathbb{R}\) one writes \(\mathrm{SO}(n)\).

It is a subgroup of the orthogonal group \(\mathrm{O}_n(R)\). If \(A\) and \(B\) lie in \(\mathrm{SO}_n(R)\), then \((AB)^{\mathsf{T}} AB = B^{\mathsf{T}} A^{\mathsf{T}} A B = I\) and \(\det(AB) = \det A \det B\) \(= 1\), so products stay in the group; it contains \(I\), and the inverse of \(A\) is \(A^{\mathsf{T}}\), which is orthogonal and has the same determinant as \(A\). Equivalently, \(\mathrm{SO}_n(R)\) is the intersection of \(\mathrm{O}_n(R)\) with the special linear group, and it is the kernel of the determinant restricted to \(\mathrm{O}_n(R)\), hence a normal subgroup of \(\mathrm{O}_n(R)\).

Over \(\mathbb{R}\), a matrix is orthogonal exactly when it preserves the dot product, \((Ax) \cdot (Ay) = x \cdot y\) for all vectors \(x, y\); the extra condition \(\det A = 1\) says that it also preserves orientation.

Examples

$\(\begin{pmatrix} a & b \\ -b & a \end{pmatrix}.\)$

Over \(\mathbb{R}\) one may write \(a = \cos\theta\) and \(b = -\sin\theta\), which gives the rotation of the plane by the angle \(\theta\). Composing rotations adds their angles, so \(\mathrm{SO}(2)\) is abelian and isomorphic to the circle group of complex numbers of modulus \(1\), the rotation by \(\theta\) corresponding to \(e^{i\theta}\). Over the field with three elements the equation \(a^2 + b^2 = 1\) has four solutions, and \(\mathrm{SO}_2\) is cyclic of order \(4\). - Dimension three. By Euler's rotation theorem, every element of \(\mathrm{SO}(3)\) is a rotation about some axis through the origin. The group is not abelian: quarter turns about two perpendicular axes do not commute.

Properties
  • Determinant. For every orthogonal matrix \(A\), \((\det A)^2 = 1\). (Mathlib proves this for unitary matrices over a commutative ring with an involution, as the statement that \(\det A\) is a unitary element, meaning that \(\det A\) times its image under the involution is \(1\); the orthogonal case is the trivial involution, see below, where this says \(\det A \cdot \det A = 1\).)
  • Index two. Over a field of characteristic other than \(2\) and for \(n \ge 1\), the determinant of an orthogonal matrix is therefore \(\pm 1\), and the diagonal matrix with entries \(-1, 1, \dots, 1\) is orthogonal of determinant \(-1\). So \(\mathrm{SO}_n\) has index \(2\) in \(\mathrm{O}_n\), the other coset consisting of the orthogonal matrices of determinant \(-1\). Over a field of characteristic \(2\), where \(-1 = 1\), every orthogonal matrix has determinant \(1\), so the two groups coincide; the group called special orthogonal in that setting is defined in another way.
  • Odd dimension. Over a field of characteristic other than \(2\) and for odd \(n\), the matrix \(-I\) is central, orthogonal and of determinant \(-1\), so \(\mathrm{O}_n\) is isomorphic to the direct product \(\mathrm{SO}_n \times \{I, -I\}\), the isomorphism being multiplication.
  • Topology. For \(n \ge 1\), \(\mathrm{SO}(n)\) is a compact, connected Lie group of dimension \(n(n-1)/2\), and it is the connected component of the identity in \(\mathrm{O}(n)\).
  • Orthonormal bases. In a finite-dimensional real inner product space, the matrix expressing one orthonormal basis in terms of another is orthogonal; its determinant is \(1\) when the two bases have the same orientation and \(-1\) when they have opposite orientations. Hence the columns of a matrix in \(\mathrm{SO}(n)\) form an orthonormal basis of \(\mathbb{R}^n\) that is positively oriented, that is, has the same orientation as the standard basis, and each such basis arises from exactly one matrix in the group.
  • The orthogonal group drops the determinant condition.
  • The special unitary group is the complex analogue; a real matrix is unitary exactly when it is orthogonal, so \(\mathrm{SO}(n)\) consists of the real matrices in \(\mathrm{SU}(n)\).
  • More generally, over a field of characteristic other than \(2\), a nondegenerate quadratic form \(Q\) has a special orthogonal group, the linear maps of determinant \(1\) that preserve \(Q\); \(\mathrm{SO}_n\) is the case of the sum of \(n\) squares.
In Mathlib

Mathlib defines the special orthogonal group for square matrices indexed by any finite type with decidable equality, with entries in any commutative ring, as a submonoid of the matrices; the inverse of an element is its transpose, which makes it a group. The definition reuses the unitary theory: a commutative ring is given the trivial involution, so the conjugate transpose becomes the transpose, the orthogonal group is the unitary group for that involution, and the special orthogonal group is the corresponding special unitary group, the unitary matrices of determinant \(1\). Mathlib's own documentation notes that this definition is the intended one only when \(2\) is invertible, matching the characteristic \(2\) caveat above.

Written for Trough, under CC BY-SA 4.0.

Sources
MathlibMatrix.specialOrthogonalGroup
Uses 3
Statement usesOrthogonal group · Determinant · Subgroup

AbbreviationUnitary group#

The unitary group \(U(n)\) is the group of \(n \times n\) complex matrices \(U\) satisfying \(U^*U = I\), where \(U^*\) is the conjugate transpose of \(U\). These are exactly the matrices of the linear maps of \(\mathbb{C}^n\) that preserve the standard Hermitian inner product, so \(U(n)\) plays for complex space the role that the orthogonal group plays for real Euclidean space. It is a compact Lie group, and it is basic to linear algebra, to the representation theory of finite and compact groups, and to quantum mechanics, where unitary matrices describe reversible changes of state.

Definition

For a complex matrix \(A\), the conjugate transpose \(A^*\) has entries \((A^*)_{ij} = \overline{A_{ji}}\), where the bar denotes conjugation of complex numbers. A square matrix \(U\) is unitary if

\[U^*U = UU^* = I,\]

that is, if \(U\) is invertible with inverse \(U^*\). The set \(U(n)\) of unitary \(n \times n\) matrices is closed under products, since \((UV)^*(UV) = V^*U^*UV = I\) and likewise \((UV)(UV)^* = UVV^*U^* = I\), and under inverses, so it is a subgroup of the general linear group \(GL_n(\mathbb{C})\). For square matrices either equation alone suffices: \(U\) is unitary exactly when \(U^*U = I\), and exactly when \(UU^* = I\).

With the inner product \(\langle x, y \rangle = \sum_k \overline{x_k}\, y_k\) on \(\mathbb{C}^n\), the \((i, j)\) entry of \(U^*U\) is the inner product of the \(i\)-th and \(j\)-th columns of \(U\). So \(U\) is unitary if and only if its columns form an orthonormal basis of \(\mathbb{C}^n\), and likewise for its rows. Equivalently, \(\langle Ux, Uy \rangle = \langle x, y \rangle\) for all vectors \(x\) and \(y\). In the language of inner product spaces, the matrix of a linear isometric equivalence with respect to orthonormal bases is unitary, and in particular the change-of-basis matrix between two orthonormal bases is unitary. Mathlib states both facts for finite-dimensional real or complex inner product spaces.

Examples
  • The circle group. A \(1 \times 1\) matrix \((z)\) is unitary when \(\overline{z}z = 1\), that is, \(|z| = 1\). So \(U(1)\) is the group of complex numbers \(e^{i\theta}\) of absolute value \(1\) under multiplication.
  • Diagonal matrices. A diagonal matrix is unitary exactly when its tuple of diagonal entries is unitary in the product ring, that is, when every diagonal entry is unitary; over \(\mathbb{C}\) this means every diagonal entry has absolute value \(1\).
  • Real matrices. A real matrix is unitary exactly when it is orthogonal, so \(U(n)\) contains the orthogonal group \(O(n)\), including all \(n \times n\) permutation matrices; for \(n = 2\) it contains the rotation matrices of the plane.
  • Pauli matrices. The matrices $\(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)$ are unitary and equal to their own conjugate transposes.
  • The group \(U(2)\). Every element of \(U(2)\) has the form $\(U = \begin{pmatrix} a & b \\ -e^{i\varphi}\,\overline{b} & e^{i\varphi}\,\overline{a} \end{pmatrix}, \qquad |a|^2 + |b|^2 = 1,\)$ for complex \(a, b\) and real \(\varphi\), and then \(\det U = e^{i\varphi}\).
Properties
Generalizations

Unitarity makes sense in any ring or monoid with a star operation, an involution that reverses products: an element \(u\) is unitary when \(u^*u = uu^* = 1\), and the unitary elements embed injectively into the group of invertible elements. For bounded operators on a Hilbert space, the adjoint plays the role of the conjugate transpose: a unitary operator preserves inner products, and the unitary operators form a group isomorphic to the group of linear isometric automorphisms of the space. In a C*-algebra, every point of the spectrum of a unitary element has absolute value \(1\). Other generalizations include the indefinite unitary groups \(U(p, q)\), which preserve a Hermitian form of signature \((p, q)\), and unitary groups over other fields with an involution, such as a finite field of order \(q^2\) with the map \(x \mapsto x^q\).

In Mathlib

Mathlib defines the unitary group for square matrices indexed by any finite type, with entries in any commutative ring carrying a star operation, an involution of the ring. The conjugate transpose is the star operation on matrices, and over the complex numbers the star operation is complex conjugation, so the classical \(U(n)\) is the case of complex entries indexed by \(n\) elements. The unitary group is not built separately: it is the set of unitary elements of the matrix ring, packaged as a submonoid of matrices, and its group structure, with inverse given by the star operation, comes from the general construction. The definition asks for both \(U^*U = I\) and \(UU^* = I\), and the one-sided criteria above show either suffices. The orthogonal group is defined as the unitary group for the trivial star operation, and the special unitary group as the unitary matrices of determinant \(1\).

Written for Trough, under CC BY-SA 4.0.

Sources
MathlibMatrix.unitaryGroup
Uses 5 · Used by 1
Statement usesRing-valued matrix · Complex number · Group · Subgroup · General linear group (Classical automorphism groups)
Used bySpecial unitary group

DefinitionSpecial unitary group#

The special unitary group \(\mathrm{SU}(n)\) is the group of complex \(n \times n\) unitary matrices whose determinant equals \(1\). It is the subgroup of the unitary group \(\mathrm{U}(n)\) cut out by the single condition \(\det A = 1\), the kernel of the determinant, and it is one of the basic examples of a compact Lie group. The group \(\mathrm{SU}(2)\) is a double cover of the rotation group \(\mathrm{SO}(3)\) of three-dimensional space, and the groups \(\mathrm{SU}(n)\) appear throughout representation theory, geometry and physics.

Definition

For a complex matrix \(A\), write \(A^* = \overline{A}^{\mathsf T}\) for its conjugate transpose. A square matrix \(A\) is unitary when \(A^*A = AA^* = I\), that is, when its inverse is \(A^*\). For square matrices over a commutative ring with an involution either equation alone suffices: \(A\) is unitary as soon as \(AA^* = I\), and likewise as soon as \(A^*A = I\). Throughout, \(n \ge 1\), and the special unitary group is

\[\mathrm{SU}(n) = \{ A \in \mathrm{U}(n) : \det A = 1 \},\]

so a matrix lies in \(\mathrm{SU}(n)\) exactly when it is unitary and has determinant \(1\). In particular \(\mathrm{SU}(n)\) is contained in \(\mathrm{U}(n)\). Equivalently, \(\mathrm{SU}(n)\) is the intersection of \(\mathrm{U}(n)\) with the special linear group \(\mathrm{SL}_n(\mathbb C)\).

It is a subgroup of \(\mathrm{U}(n)\). Since \(\det(AB) = \det A \det B\), a product of two matrices of determinant \(1\) again has determinant \(1\). The inverse of a unitary matrix is its conjugate transpose, and \(\det A^* = \overline{\det A}\), so the inverse of an element of \(\mathrm{SU}(n)\) is its conjugate transpose and again lies in \(\mathrm{SU}(n)\).

Examples
  • Dimension one. \(\mathrm{U}(1)\) is the unit circle \(\{ z \in \mathbb C : |z| = 1 \}\), and \(\mathrm{SU}(1)\) is the trivial group.
  • Dimension two. Every element of \(\mathrm{SU}(2)\) has the form

$\(\begin{pmatrix} a & -\overline{b} \\ b & \overline{a} \end{pmatrix}, \qquad a, b \in \mathbb C, \quad |a|^2 + |b|^2 = 1,\)$

and every such matrix lies in \(\mathrm{SU}(2)\). As a space, \(\mathrm{SU}(2)\) is therefore the unit sphere \(S^3\) in \(\mathbb C^2 \cong \mathbb R^4\), and as a group it is isomorphic to the group of unit quaternions. It maps onto the rotation group \(\mathrm{SO}(3)\) by a two-to-one homomorphism with kernel \(\{ \pm I \}\). - Diagonal matrices. The diagonal matrices \(\operatorname{diag}(e^{i\theta_1}, \dots, e^{i\theta_n})\) with \(\theta_1 + \dots + \theta_n \in 2\pi\mathbb Z\) form an abelian subgroup of \(\mathrm{SU}(n)\), a torus of dimension \(n - 1\). - Real matrices. A real matrix in \(\mathrm{SU}(n)\) is an orthogonal matrix of determinant \(1\), so the special orthogonal group \(\mathrm{SO}(n)\) is a subgroup of \(\mathrm{SU}(n)\).

Properties
  • The determinant of a unitary matrix. For \(A\) unitary, \(\det A\) is unitary as an element of the ring, that is, \(\overline{\det A}\,\det A = \det A\,\overline{\det A} = 1\); over \(\mathbb C\) this says \(|\det A| = 1\). The determinant is therefore a group homomorphism from \(\mathrm{U}(n)\) onto \(\mathrm{U}(1)\), surjective because \(\operatorname{diag}(z, 1, \dots, 1)\) has determinant \(z\). Its kernel is \(\mathrm{SU}(n)\), so \(\mathrm{SU}(n)\) is a normal subgroup of \(\mathrm{U}(n)\) with quotient \(\mathrm{U}(1)\).
  • Splitting. Every unitary matrix \(U\) factors uniquely as \(U = D S\) with \(D = \operatorname{diag}(\det U, 1, \dots, 1)\) and \(S \in \mathrm{SU}(n)\), so \(\mathrm{U}(n)\) is a semidirect product of \(\mathrm{SU}(n)\) and \(\mathrm{U}(1)\). The multiplication map \(\mathrm{U}(1) \times \mathrm{SU}(n) \to \mathrm{U}(n)\), \((\lambda, S) \mapsto \lambda S\), is a surjective homomorphism whose kernel is cyclic of order \(n\).
  • Centre. The centre of \(\mathrm{SU}(n)\) consists of the scalar matrices \(\omega I\) with \(\omega^n = 1\), a cyclic group of order \(n\).
  • Diagonalization. Every element of \(\mathrm{SU}(n)\) is conjugate, within \(\mathrm{SU}(n)\), to a diagonal matrix whose entries have absolute value \(1\) and product \(1\).
  • Topology. \(\mathrm{SU}(n)\) is a compact, connected and simply connected real Lie group of dimension \(n^2 - 1\). Its Lie algebra \(\mathfrak{su}(n)\) consists of the skew-Hermitian matrices of trace \(0\).
Generalizations

The definition makes sense over any commutative ring \(R\) with an involution \(a \mapsto \overline{a}\): the conjugate transpose is formed with that involution, and the special unitary group consists of the matrices with \(A^*A = AA^* = I\) and determinant \(1\). Taking the field with \(q^2\) elements, with the involution \(a \mapsto a^q\), gives the finite special unitary groups. When \(2\) is invertible in \(R\), taking the trivial involution gives the special orthogonal group, since then \(A^*\) is just the transpose; in characteristic \(2\) the matrices with \(A^{\mathsf T}A = I\) and determinant \(1\) do not form the usual special orthogonal group of a quadratic form. Related matrix groups include the general linear group and the orthogonal group.

In Mathlib

Mathlib defines the special unitary group in this general form: for any finite index type and any commutative ring with an involution, it is the set of square matrices that are unitary and have determinant \(1\). On matrices the involution is the conjugate transpose, and unitary means that both products with the conjugate transpose are the identity. The set is built as a submonoid of all square matrices, namely the intersection of the unitary group with the kernel of the determinant, viewed as a homomorphism of multiplicative monoids; the group structure is then added, with the conjugate transpose as inverse. The special orthogonal group is the same construction with the trivial involution. The documentation of the definition notes that it is only correct when \(2\) is invertible, the same restriction as for the trivial involution described under Generalizations. Mathlib also allows an empty index type, for which the group is trivial.

Written for Trough, under CC BY-SA 4.0.

Sources
MathlibMatrix.specialUnitaryGroup
Uses 4
Statement usesUnitary group · Determinant · Complex number · Subgroup

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