Orthogonal group

In mathematics, the orthogonal group or rotation group is the group of distance-preserving transformations of Euclidean space which preserve the origin, where the group operation is given by composing transformations. Equivalently, it is the group of orthogonal matrices of a given dimension, where the group operation is given by matrix multiplication. An orthogonal matrix is a real matrix whose inverse equals its transpose. The term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators which preserve a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field.

In particular, when the bilinear form is the scalar product on the vector space Fⁿ of dimension n over a field F, with quadratic form the sum of squares, then the corresponding orthogonal group, written as O(n, F ), is the set of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication. This is a subgroup of the general linear group GL(n, F ) given by

$\mathrm {O} (n,F)=\{Q\in \mathrm {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\}$

where Q^T is the transpose of Q and I is the identity matrix. The classical orthogonal group over the real numbers is usually just written O(n). This characterization can be derived as follows: let b be a bilinear form, and let $Q_{i}^{j}$ be the components of a linear transformation with respect to some basis {e_i} . We may, in fact, choose {e_i} to be such that b(e_i, e_j) is 1 if i = j and 0 otherwise, using the Gram–Schmidt process. Then in order for Q to preserve b, for any e_i, e_j, we must have

${\begin{aligned}b(e_{i},e_{j})&=b(Q(e_{i}),Q(e_{j}))\\&=b(\sum _{k}Q_{i}^{k}e_{k},\sum _{n}Q_{j}^{n}e_{n})\\&=\sum _{n,k}Q_{i}^{k}Q_{j}^{n}b(e_{k},e_{n})\\&=\sum _{k}Q_{i}^{k}Q_{j}^{k}\end{aligned}}$

The final term is just the (i, j) component of Q Q^T, which must therefore be 1 on the diagonal and 0 off it. This implies that Q Q^T = I. Conversely, if Q preserves the form for basis vectors, it is easy to check that by linearity, Q preserves it for all vectors.

This article mainly discusses definite forms: the orthogonal group of the positive definite form (equivalent to the sum of n squares). Negative definite forms (equivalent to the negative sum of n squares) are identical since O(n, 0) = O(0, n). However, the associated Pin groups differ.

For O(p, q) of other non-singular forms, see indefinite orthogonal group.

The determinant of any orthogonal matrix is either 1 or −1. The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n, F ) known as the special orthogonal group SO(n, F ). (More precisely, SO(n, F ) is the kernel of the Dickson invariant, discussed below.) By analogy with GL–SL (general linear group, special linear group), the orthogonal group is sometimes called the general orthogonal group and denoted GO, though this term is also sometimes used for indefinite orthogonal groups O(p, q).

The derived subgroup Ω(n, F ) of O(n, F) is an often studied object because, when F is a finite field, Ω(n, F ) is often a central extension of a finite simple group.

Both O(n, F ) and SO(n, F ) are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix. The Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form.

Even and odd dimension

The structure of the orthogonal group differs in certain aspects between even and odd dimensions – for example, over ordered fields (such as R) the −I element is orientation-preserving in even dimensions, but orientation-reversing in odd dimensions. When this distinction wishes to be emphasized, the groups are generally denoted O(2k) and O(2k + 1), reserving n for the dimension of the space (n = 2k or n = 2k + 1). The letters p or r are also used, indicating the rank of the corresponding Lie algebra; in odd dimension the corresponding Lie algebra is ${\mathfrak {so}}(2r+1)$, while in even dimension the Lie algebra is ${\mathfrak {so}}(2r)$.

Over the real number field

Over the field R of real numbers, the orthogonal group O(n, R) and the special orthogonal group SO(n, R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n − 1)/2. O(n, R) has two connected components, with SO(n, R) being the identity component, i.e., the connected component containing the identity matrix.

Geometric interpretation

The real orthogonal and real special orthogonal groups have the following geometric interpretations:

O(n, R) is a subgroup of the Euclidean group E(n), the group of isometries of Rⁿ; it contains those that leave the origin fixed – O(n, R) = E(n) ∩ GL(n, R). It is the symmetry group of the sphere (n = 3) or (n − 1)-sphere and all objects with spherical symmetry, if the origin is chosen at the center.

SO(n, R) is a subgroup of E⁺(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed – SO(n, R) = E⁺(n) ∩ GL(n, R) = E(n) ∩ GL⁺(n, R). It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.

{±I} is a normal subgroup and even a characteristic subgroup of O(n, R), and, if n is even, also of SO(n, R). If n is odd, O(n, R) is the internal direct product of SO(n, R) and {±I}. For every positive integer k the cyclic group C_k of k-fold rotations is a normal subgroup of O(2, R) and SO(2, R).

Relative to suitable orthogonal bases, the isometries are of the form:

${\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}}$

where the matrices R₁, … , R_k are 2-by-2 rotation matrices in orthogonal planes of rotation. As a special case, known as Euler's rotation theorem, any (non-identity) element of SO(3, R) is rotation about a uniquely defined axis.

The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group, and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).

The symmetry group of a circle is O(2, R). It is isomorphic (as a real Lie group) to the circle group, also known as U(1). This isomorphism sends the complex number exp(φ i) = cos φ + i sin φ of absolute value 1 to the orthogonal matrix

${\begin{bmatrix}\cos(\phi )&-\sin(\phi )\\\sin(\phi )&\cos(\phi )\end{bmatrix}}.$

The group SO(3, R), understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering, and there are numerous charts on SO(3).

Low-dimensional topology

The low dimensional (real) orthogonal groups are familiar spaces:

• O(1) = {±1}, a two-point discrete space
• SO(1) = {1}
• SO(2) is S¹
• SO(3) is RP³
• SO(4) is double covered by SU(2) × SU(2) = S³ × S³.

Homotopy groups

In terms of algebraic topology, for n > 2 the fundamental group of SO(n, R) is cyclic of order 2, and the spin group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group Spin(2) is the unique 2-fold cover).

Generally, the homotopy groups π_k(O) of the real orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions:

$\mathrm {O} (0)\subset \mathrm {O} (1)\subset \mathrm {O} (2)\subset \cdots \subset O=\bigcup _{k=0}^{\infty }\mathrm {O} (k)$

Since the inclusions are all closed, hence cofibrations, this can also be interpreted as a union. On the other hand Sⁿ is a homogeneous space for O(n + 1), and one has the following fiber bundle:

$\mathrm {O} (n)\to \mathrm {O} (n+1)\to S^{n},$

which can be understood as "The orthogonal group O(n + 1) acts transitively on the unit sphere Sⁿ, and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower. Thus the natural inclusion O(n) → O(n + 1) is (n − 1)-connected, so the homotopy groups stabilize, and π_k( π_k(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

From Bott periodicity we obtain Ω⁸O ≅ O, therefore the homotopy groups of O are 8-fold periodic, meaning π_k + 8(O) = π_k(O), and one needs only to compute the lower 8 homotopy groups:

${\begin{aligned}\pi _{0}(O)&=\mathbf {Z} /2\\\pi _{1}(O)&=\mathbf {Z} /2\\\pi _{2}(O)&=0\\\pi _{3}(O)&=\mathbf {Z} \\\pi _{4}(O)&=0\\\pi _{5}(O)&=0\\\pi _{6}(O)&=0\\\pi _{7}(O)&=\mathbf {Z} \end{aligned}}$

Relation to KO-theory

Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: π_k(O) = π_k + 1(BO). Setting KO = BO × Z = Ω⁻¹O × Z (to make π₀ fit into the periodicity), one obtains:

${\begin{aligned}\pi _{0}(KO)&=\mathbf {Z} \\\pi _{1}(KO)&=\mathbf {Z} /2\\\pi _{2}(KO)&=\mathbf {Z} /2\\\pi _{3}(KO)&=0\\\pi _{4}(KO)&=\mathbf {Z} \\\pi _{5}(KO)&=0\\\pi _{6}(KO)&=0\\\pi _{7}(KO)&=0\end{aligned}}$

Computation and interpretation of homotopy groups

Low-dimensional groups

The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.

• π₀(O) = π₀(O(1)) = Z/2, from orientation-preserving/reversing (this class survives to O(2) and hence stably)
• π₁(O) = π₁(SO(3)) = Z/2, which is spin comes from SO(3) = RP³ = S³/(Z/2).
• π₂(O) = π₂(SO(3)) = 0, which surjects onto π₂(SO(4)); this latter thus vanishes.

Lie groups

From general facts about Lie groups, π₂(G) always vanishes, and π₃(G) is free (free abelian).

Vector bundles

From the vector bundle point of view, π₀(KO) is vector bundles over S⁰, which is two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so π₀(KO) = Z is dimension.

Loop spaces

Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret higher homotopy of O as lower homotopy of simple to analyze spaces. Using π₀, O and O/U have two components, KO = BO × Z and KSp = BSp × Z have countably many components, and the rest are connected.

Interpretation of homotopy groups

In a nutshell:

• π₀(KO) = Z is about dimension
• π₁(KO) = Z/2 is about orientation
• π₂(KO) = Z/2 is about spin
• π₄(KO) = Z is about topological quantum field theory.

Let R be any of four division rings R, C, H, O, and let L_R be the tautological line bundle over the projective line RP¹, and [L_R] its class in K-theory. Noting that RP¹ = S¹, CP¹ = S², HP¹ = S⁴, OP¹ = S⁸, these yield vector bundles over the corresponding spheres, and

• π₁(KO) is generated by [L_R]
• π₂(KO) is generated by [L_C]
• π₄(KO) is generated by [L_H]
• π₈(KO) is generated by [L_O]

From the point of view of symplectic geometry, π₀(KO) ≅ π₈(KO) = Z can be interpreted as the Maslov index, thinking of it as the fundamental group π₁(U/O) of the stable Lagrangian Grassmannian as U/O ≅ Ω⁷(KO), so π₁(U/O) = π_1 + 7(KO).

Over the complex number field

Over the field C of complex numbers, O(n, C) and SO(n, C) are complex Lie groups of dimension n(n − 1)/2 over C (it means the dimension over R is twice that). O(n, C) has two connected components, and SO(n, C) is the connected component containing the identity matrix. For n ≥ 2 these groups are noncompact.

Just as in the real case SO(n, C) is not simply connected. For n > 2 the fundamental group of SO(n, C) is cyclic of order 2 whereas the fundamental group of SO(2, C) is infinite cyclic.

Over finite fields

Orthogonal groups can also be defined over finite fields F_q, where q is a power of a prime p. When defined over such fields, they come in two types in even dimension: O⁺(2n, q) and O⁻(2n, q); and one type in odd dimension: O(2n + 1, q).

If V is the vector space on which the orthogonal group G acts, it can be written as a direct orthogonal sum as follows:

$V=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{m}\oplus W,$

where L_i are hyperbolic lines and W contains no singular vectors. If W is the zero subspace, then G is of plus type. If W is one-dimensional then G has odd dimension. If W has dimension 2, G is of minus type.

In the special case where n = 1, O^ϵ(2, q) is a dihedral group of order 2(q − ϵ).

We have the following formulas for the order of O(n, q), when the characteristic is greater than two:

$|\mathrm {O} (2n+1,q)|=2q^{n}\prod _{i=0}^{n-1}(q^{2n}-q^{2i}).$

If −1 is a square in F_q

$|\mathrm {O} (2n,q)|=2(q^{n}-1)\prod _{i=1}^{n-1}(q^{2n}-q^{2i}).$

If −1 is a non-square in F_q

$|\mathrm {O} (2n,q)|=2(q^{n}+(-1)^{n+1})\prod _{i=1}^{n-1}(q^{2n}-q^{2i}).$

The Dickson invariant

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group Z/2Z (integers modulo 2), taking the value 0 in case the element is the product of an even number of reflections, and the value of 1 otherwise.

Algebraically, the Dickson invariant can be defined as D(f) = rank(I − f) modulo 2, where I is the identity (Taylor 1992, Theorem 11.43). Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.

The special orthogonal group is the kernel of the Dickson invariant and usually has index 2 in O(n, F ). When the characteristic of F is not 2, the Dickson Invariant is 0 whenever the determinant is 1. Thus when the characteristic is not 2, SO(n, F ) is commonly defined to be the elements of O(n, F ) with determinant 1. Each element in O(n, F ) has determinant ±1. Thus in characteristic 2, the determinant is always 1.

The Dickson invariant can also be defined for Clifford groups and Pin groups in a similar way (in all dimensions).

Orthogonal groups of characteristic 2

Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section. (Formerly these groups were known as the hypoabelian groups but this term is no longer used.)

• Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4 dimensional over the field with 2 elements and the Witt index is 2. Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector u takes a vector v to v + B(v, u)/Q(u) · u where B is the bilinear form and Q is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes v to v − 2·B(v, u)/Q(u) · u.

• The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since I = −I.

• In odd dimensions 2n + 1 in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension 2n, acted upon by the orthogonal group.

• In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.

The spinor norm

The spinor norm is a homomorphism from an orthogonal group over a field F to the quotient group F */F *² (the multiplicative group of the field F up to square elements), that takes reflection in a vector of norm n to the image of n in F */F *².

For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.

Galois cohomology and orthogonal groups

In the theory of Galois cohomology of algebraic groups, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomena is concerned. The first point is that quadratic forms over a field can be identified as a Galois H¹, or twisted forms (torsors) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.

The 'spin' name of the spinor norm can be explained by a connection to the spin group (more accurately a pin group). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras). The spin covering of the orthogonal group provides a short exact sequence of algebraic groups.

$1\rightarrow \mu _{2}\rightarrow \mathrm {Pin} _{V}\rightarrow \mathrm {O_{V}} \rightarrow 1$

Here μ₂ is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The connecting homomorphism from H⁰(O_V), which is simply the group O_V(F ) of F-valued points, to H¹(μ₂) is essentially the spinor norm, because H¹(μ₂) is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from H¹ of the orthogonal group, to the H² of the kernel of the spin covering. The cohomology is non-abelian, so that this is as far as we can go, at least with the conventional definitions.

Lie algebra

The Lie algebra corresponding to Lie groups O(n, F ) and SO(n, F ) consists of the skew-symmetric n × n matrices, with the Lie bracket [ , ] given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by ${\mathfrak {o}}(n,F)$ or ${\mathfrak {so}}(n,F)$, and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different n are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension B_k, where n = 2k + 1, while in even dimension D_r, where n = 2r.

More intrinsically, given a vector space with an inner product, the special orthogonal Lie algebra is given by the bivectors on the space, which are sums of simple bivectors (2-blades) v ∧ w. The correspondence is given by the map $\mathbf {v} \wedge \mathbf {w} \mapsto \mathbf {v} ^{*}\otimes \mathbf {w} -\mathbf {w} ^{*}\otimes \mathbf {v} ,$ where v* is the covector dual to the vector v; in coordinates these are exactly the elementary skew-symmetric matrices.

Over real numbers, this characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name. Generalizing the inner product with a nondegenerate form yields the indefinite orthogonal Lie algebras ${\mathfrak {so}}(p,q).$

The representation theory of the orthogonal Lie algebras includes both representations corresponding to linear representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups (linear representations of spin groups), the so-called spin representation, which are important in physics.

Related groups

The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.

The inclusions O(n) ⊂ U(n) ⊂ Sp(n) = USp(2n) and USp(n) ⊂ U(n) ⊂ O(2n) are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, U(n)/O(n) is the Lagrangian Grassmannian.

Lie subgroups

In physics, particularly in the areas of Kaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:

$\mathrm {O} (n)\supset \mathrm {O} (n-1)$ – preserve an axis

$\mathrm {O} (2n)\supset \mathrm {U} (n)\supset \mathrm {SU} (n)$ – U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; SU(n) also preserves a complex orientation.

$\mathrm {O} (2n)\supset \mathrm {USp} (n)$

$\mathrm {O} (7)\supset \mathbf {G} _{2}$

Lie supergroups

The orthogonal group O(n) is also an important subgroup of various Lie groups:

$\mathrm {U} (n)\supset \mathrm {SU} (n)\supset \mathrm {O} (n)$

$\mathrm {USp} (2n)\supset \mathrm {O} (n)$

$\mathbf {G} _{2}\supset \mathrm {O} (3)$

$\mathbf {F} _{4}\supset \mathrm {O} (9)$

$\mathbf {E} _{6}\supset \mathrm {O} (10)$

$\mathbf {E} _{7}\supset \mathrm {O} (12)$

$\mathbf {E} _{8}\supset \mathrm {O} (16)$

Conformal group

Main article: Conformal group

Being isometries, real orthogonal transforms preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (Side-Side-Side) congruence of triangles and AAA (Angle-Angle-Angle) similarity of triangles. The group of conformal linear maps of Rⁿ is denoted CO(n) for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If n is odd, these two subgroups do not intersect, and they are a direct product: CO(2k + 1) = O(2k + 1) × R*, where R* = R\{0} is the real multiplicative group, while if n is even, these subgroups intersect in ±1, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: CO(2k) = O(2k) × R⁺.

Similarly one can define CSO(n); note that this is always: CSO(n) = CO(n) ∩ GL⁺(n) = SO(n) × R⁺.

Discrete subgroups

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. These subgroups are known as point group and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes.

Dimension 3 is particularly studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – see point groups in two dimensions.

Other finite subgroups include:

• Permutation matrices (the Coxeter group A_n)
• Signed permutation matrices (the Coxeter group B_n); also equals the intersection of the orthogonal group with the integer matrices.

Covering and quotient groups

The orthogonal group is neither simply connected nor centerless, and thus has both a covering group and a quotient group, respectively:

• Two covering Pin groups, Pin₊(n) → O(n) and Pin₋(n) → O(n),
• The quotient projective orthogonal group, O(n) → PO(n).

These are all 2-to-1 covers.

For the special orthogonal group, the corresponding groups are:

• Spin group, Spin(n) → SO(n),
• Projective special orthogonal group, SO(n) → PSO(n).

Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, and in odd dimension PSO(2k + 1) is a 1-to-1 cover, i.e., isomorphic to SO(2k + 1). These groups, Spin(n), SO(n), and PSO(n) are Lie group forms of the compact special orthogonal Lie algebra, ${\mathfrak {so}}(n,{\mathbb {R} })$ – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.

In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.

Principal homogeneous space: Stiefel manifold

Main article: Stiefel manifold

The principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold V_n(Rⁿ) of orthonormal bases (orthonormal n-frames).

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

The other Stiefel manifolds V_k(Rⁿ) for k < n of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.

See also

Specific transforms

• Coordinate rotations and reflections
• Reflection through the origin

Specific groups

• rotation group, SO(3, R)
• SO(8)

Related groups

• indefinite orthogonal group
• unitary group
• symplectic group

Lists of groups

• list of finite simple groups
• list of simple Lie groups

References

• Cassels, J.W.S. (1978), Rational Quadratic Forms, London Mathematical Society Monographs, vol. 13, Academic Press, ISBN 0-12-163260-1, Zbl 0395.10029
• Grove, Larry C. (2002), Classical groups and geometric algebra, Graduate Studies in Mathematics, vol. 39, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2019-3, MR 1859189
• Taylor, Donald E. (1992), The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, vol. 9, Berlin: Heldermann Verlag, ISBN 3-88538-009-9, MR 1189139, Zbl 0767.20001

External links

• "Orthogonal group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• John Baez "This Week's Finds in Mathematical Physics" week 105
• John Baez on Octonions
• Template:It n-dimensional Special Orthogonal Group parametrization