A square matrix can be invertible without respecting the geometry of the space it acts on. There are linear maps of determinant $1$ that preserve oriented volume in $\mathbb{R}^n$ but bend right angles into slanted angles. The orthogonal group is designed to separate genuine Euclidean symmetries from merely invertible or volume-preserving linear maps.

From the point of view of [group theory](/page/Group), the orthogonal group is a child of the general notion of a group: its operation is matrix multiplication, its identity is the identity matrix, and its inverses are matrix inverses. What makes it special is the additional geometric constraint that the Euclidean [inner product](/page/Inner%20Product) must be preserved.

The first failure is a shear. It has determinant $1$, so a determinant-only test would treat it like a rotation. Looking at two perpendicular vectors shows why that test is too weak.

[example: Area Preservation Does Not Preserve Angles] Let $A:\mathbb{R}^2\to\mathbb{R}^2$ be the [linear map](/page/Linear%20Map) determined by $Ae_1=e_1$ and $Ae_2=e_1+e_2$, where $e_1=(1,0)$ and $e_2=(0,1)$. Since the columns of the standard matrix of a linear map are the images of the standard basis vectors, the first column of $A$ is $Ae_1=(1,0)$ and the second column is $Ae_2=e_1+e_2=(1,0)+(0,1)=(1,1)$. Hence \begin{align*} A=\begin{pmatrix}1&1\cr 0&1\end{pmatrix}. \end{align*} For a $2\times 2$ matrix $\begin{pmatrix}a&b\cr c&d\end{pmatrix}$, the determinant is $ad-bc$. Therefore \begin{align*} \det A=(1)(1)-(1)(0)=1-0=1. \end{align*} So this shear preserves signed area. Now compare what $A$ does to the standard perpendicular basis vectors. Their original Euclidean inner product is \begin{align*} \langle e_1,e_2\rangle=\langle (1,0),(0,1)\rangle=(1)(0)+(0)(1)=0+0=0. \end{align*} The images are \begin{align*} Ae_1=e_1=(1,0). \end{align*} Also, \begin{align*} Ae_2=e_1+e_2=(1,0)+(0,1)=(1+0,0+1)=(1,1). \end{align*} Their new inner product is \begin{align*} \langle Ae_1,Ae_2\rangle=\langle (1,0),(1,1)\rangle=(1)(1)+(0)(1)=1+0=1. \end{align*} Thus two input vectors with inner product $0$ are sent to two output vectors with inner product $1$, so perpendicularity is not preserved. The same map changes the length of $e_2$. Since $Ae_2=(1,1)$, its squared length is \begin{align*} |Ae_2|^2=\langle Ae_2,Ae_2\rangle=\langle (1,1),(1,1)\rangle=(1)(1)+(1)(1)=1+1=2. \end{align*} Meanwhile, \begin{align*} |e_2|^2=\langle e_2,e_2\rangle=\langle (0,1),(0,1)\rangle=(0)(0)+(1)(1)=0+1=1. \end{align*} Since $|Ae_2|^2=2$ and $|e_2|^2=1$, the vector $e_2$ does not keep its length under $A$. Determinant $1$ records signed area preservation here, but it does not force preservation of lengths or angles; the missing condition is preservation of the Euclidean inner product. [/example]

This page develops the matrix equation that rules out this failure. We start with the definition, then translate it into preservation of lengths and angles, separate rotations from reflections by determinant, study the action on spheres, and finish with the infinitesimal [Lie algebra](/page/Lie%20Algebra) viewpoint.

## Definition

The shear example shows that determinant alone does not recognise Euclidean geometry. The missing datum is not area but the whole dot-product structure: a genuine linear symmetry must preserve the Euclidean inner product between every pair of vectors. The primary definition therefore builds the group directly from that preservation law.

[definition: Orthogonal Group] For $n\in\mathbb{N}$, let $M_n(\mathbb{R})$ denote the set of real $n\times n$ matrices, and let $GL(n,\mathbb{R})$ denote the group of invertible matrices in $M_n(\mathbb{R})$. For a matrix $A$, let $A^\top$ denote its transpose, and let $I_n$ denote the $n\times n$ identity matrix. The orthogonal group of degree $n$ over $\mathbb{R}$ is the group \begin{align*} O(n)=O_n(\mathbb{R}):=\{A\in GL(n,\mathbb{R}):A^\top A=I_n\}, \end{align*} with group operation $m:O(n)\times O(n)\to O(n)$ given by \begin{align*} m(A,B)=AB. \end{align*} [/definition]

The inverse of an element of $O(n)$ is its transpose, so the definition is rigid enough to make composition and reversal stay inside the same class of symmetries. The rest of the page explains why the equation $A^\top A=I_n$ is exactly the algebraic form of preserving lengths, angles, orthonormal bases, and orientation data.

## Inner Products and Lengths

### The Euclidean Inner Product

The defining equation of $O(n)$ becomes geometric only after naming the pairing it preserves. In Euclidean space, that pairing is the dot product: it measures length when both inputs agree and angle when they do not.

[definition: Euclidean Inner Product] For $n \in \mathbb{N}$, the Euclidean inner product on $\mathbb{R}^n$ is the map $\langle \cdot,\cdot\rangle:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ defined by \begin{align*} \langle x,y\rangle=\sum_{i=1}^n x_i y_i. \end{align*} [/definition]

The associated Euclidean norm is $|x|=\sqrt{\langle x,x\rangle}$. This norm statement is not an extra axiom; it is the standard way the inner product produces lengths.

Requiring a linear map to preserve every dot product appears to be an infinite family of conditions, one for each pair of vectors. Matrix multiplication gives a compact way to express exactly that obstruction: the columns must interact with dot products as the standard basis does.

[definition: Orthogonal Matrix] Let $n \in \mathbb{N}$. Write $M_n(\mathbb{R})$ for the set of all real $n\times n$ matrices. A matrix $A\in M_n(\mathbb{R})$ is orthogonal if \begin{align*} A^\top A=I_n. \end{align*} [/definition]

### Preservation Laws

The defining equation of $O(n)$ is useful because it packages several geometric properties into one algebraic condition. The next theorem is needed to justify the definition: it proves that $A^\top A=I_n$ is not an arbitrary equation but exactly the preservation of the Euclidean inner product.

[quotetheorem:8281]

This theorem turns a matrix condition into a geometric preservation law. The next natural question is whether it is enough to test only lengths rather than all pairwise inner products, since lengths are often easier to see.