Orthogonal group

Intuition [Beginner]

The orthogonal group is the collection of transformations that preserve lengths and right angles. In the plane, it contains rotations and reflections. In three-dimensional space, it contains ordinary rotations, mirror reflections, and combinations of those.

The key point is preservation. A square may turn or flip, but its side lengths and corner angles stay the same.

The special orthogonal group keeps only the orientation-preserving part. In three dimensions, that is the group of rotations.

Visual [Beginner]

An orthogonal transformation moves a frame while keeping the two axes perpendicular and the same length.

Rotations preserve orientation; reflections reverse it.

Worked example [Beginner]

In the plane, rotate every vector by $90$ degrees. The vector pointing east moves north, and the vector pointing north moves west.

Both vectors still have length $1$, and they still meet at a right angle. So this rotation is an orthogonal transformation.

Reflecting across the horizontal axis also preserves lengths and right angles. It is orthogonal too, but it reverses orientation.

What this tells us: orthogonal transformations are the rigid linear symmetries of Euclidean space.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $(V,⟨-,-⟩)$ be a finite-dimensional real inner-product space. The orthogonal group is

$$\mathrm{O}(V)=\{A\in \mathrm{GL}(V):⟨Av,Aw⟩=⟨v,w⟩\text{for all }v,w\in V\}.$$

For $V={\mathbb{R}}^n$ with the standard inner product, this is $\mathrm{O}(n)$. In matrices,

$$\mathrm{O}(n)=\{A\in {\mathrm{GL}}_n(\mathbb{R}):A^{T}A=I\}.$$

The special orthogonal group is $\mathrm{SO}(n)=\{A\in \mathrm{O}(n):\det A=1\}$ ^{[Hall Ch. 1]}.

Key theorem with proof [Intermediate+]

Theorem (Orthogonal matrices have determinant $\pm 1$). If $A\in \mathrm{O}(n)$, then $\det A=\pm 1$.

Proof. The defining equation is $A^{T}A=I$. Taking determinants gives

$$\det (A^{T}A)=\det I.$$

Since $\det (A^T)=\det A$, the left side is $(\det A{)}^2$, and the right side is $1$. Thus $(\det A{)}^2=1$, so $\det A=\pm 1$. $\square$

Bridge. The construction here builds toward 03.05.03 (orthogonal frame bundle), where the same data is upgraded, and the symmetry side is taken up in 03.09.03 (spin group). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none is recorded because this unit needs a local choice between matrix orthogonal groups and coordinate-free linear isometries before it should be tied to Mathlib.

Advanced results [Master]

The group $\mathrm{O}(n)$ is a compact Lie subgroup of ${\mathrm{GL}}_n(\mathbb{R})$ cut out by the equation $A^{T}A=I$. Its identity component is $\mathrm{SO}(n)$, and the determinant map $\mathrm{O}(n)\to \{\pm 1\}$ records the two components ^{[Hall Ch. 1]}.

The Lie algebra is $\mathfrak{so}(n)=\{X:X^T+X=0\}$. For a real quadratic form of signature $(p,q)$, the same construction gives $\mathrm{O}(p,q)$, the group preserving that form 01.01.15.

Synthesis. This construction generalises the pattern fixed in 01.02.01 (group), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

The subgroup property follows from the defining equation: if $A^{T}A=I$ and $B^{T}B=I$, then $(AB{)}^T(AB)=B^T{A}^{T}AB=B^{T}B=I$, and $(A^{-1}{)}^T{A}^{-1}=I$ follows from $A^{-1}=A^T$.

Differentiating a smooth path $A(t)\in \mathrm{O}(n)$ with $A(0)=I$ gives $A^{\prime }(0{)}^T+A^{\prime }(0)=0$. Conversely, skew-symmetric matrices exponentiate to orthogonal matrices because $\exp (tX{)}^T\exp (tX)=\exp (-tX)\exp (tX)=I$.

Connections [Master]

• Orthogonal groups depend on groups 01.02.01, bilinear forms 01.01.15, and Lie groups 03.03.01. They act on orthonormal frames, giving the structure group of the orthogonal frame bundle 03.05.03. The spin group 03.09.03 is a double cover of $\mathrm{SO}(n)$ built from Clifford algebra 03.09.02.

• Orthogonal groups also appear in characteristic classes 03.06.04, where real vector bundles have transition functions reducible to $\mathrm{O}(n)$ after a metric is chosen.

Historical & philosophical context [Master]

Orthogonal groups arise from the classical study of quadratic forms and Euclidean geometry. Lang treats them as automorphism groups of bilinear forms ^{[Lang §XV]}.

In Lie theory, $\mathrm{O}(n)$ and $\mathrm{SO}(n)$ are among the basic matrix Lie groups; Hall introduces them as central examples of closed subgroups with explicit Lie algebras ^{[Hall Ch. 1]}.

Bibliography [Master]

• Serge Lang, Algebra, §XV. ^{[Lang §XV]}
• Brian Hall, Lie Groups, Lie Algebras, and Representations, Ch. 1. ^{[Hall Ch. 1]}
• Michael Artin, Algebra, Ch. 6, for symmetry examples. ^{[Artin Ch. 6]}