Euclidean vector space/Isometry/Section

We now discuss isometries of a Euclidean vector space in itself. These are always stets bijective. With respect to any orthonormal basis of ${}V$, they are described in the following way.

Lemma

Let ${}V$ be a Euclidean vector space, and let ${}u_{1},\ldots ,u_{n}$ denote an orthonormal basis of ${}V$. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping, and let ${}M$ be the describing matrix of ${}\varphi$ with respect to the given basis. Then ${}\varphi$ is an isometry if and only if

${}{M^{\text{tr}}}M=E_{n}\,$

holds.

Proof

Suppose first that ${}\varphi$ is an isometry. Then, ${}v_{i}=\varphi (u_{i})$ is an orthonormal basis due to fact. The coordinates of ${}v_{i}$ with respect to ${}u_{i}$ constitute the columns of the describing matrix ${}M$. Therefore, using exercise, we have

${}{v_{i}^{\text{tr}}}v_{j}=\left\langle v_{i},v_{j}\right\rangle =\delta _{ij}\,.$

Read as a matrix equation, this means

${}{M^{\text{tr}}}M={\begin{pmatrix}1&0&\cdots &\cdots &0\\0&1&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&1&0\\0&\cdots &\cdots &0&1\end{pmatrix}}\,.$

This argument can be read backwards to get the reverse implication.

$\Box$

The set of isometries on a Euclidean vector space form a group; in fact, it is a subgroup of the group of all bijective linear mappings. We recall briefly the corresponding definitions.

For a field ${}K$ and ${}n\in \mathbb {N} _{+}$, the set of all invertible ${}n\times n$-matrices with entries in ${}K$ is called the general linear group

over ${}K$. It is denoted by ${}\operatorname {GL} _{n}\!{\left(K\right)}$.

For a field ${}K$, and ${}n\in \mathbb {N} _{+}$, the set of all invertible ${}n\times n$-matrices over ${}K$ with

${}\det M=1\,$

is called the special linear group

over ${}K$. It is denoted by ${}\operatorname {SL} _{n}\!{\left(K\right)}$.

Definition

Let ${}K$ be a field, and ${}E_{n}$ the identity matrix of length ${}n$. A matrix ${}M\in \operatorname {GL} _{n}\!{\left(K\right)}$ fulfilling

${}{M^{\text{tr}}}M=E_{n}\,$

is called an orthogonal matrix. The set of all orthogonal matrices is called orthogonal group; it is denoted by

${}\operatorname {O} _{n}\!{\left(K\right)}={\left\{M\in \operatorname {GL} _{n}\!{\left(K\right)}\mid {M^{\text{tr}}}M=E_{n}\right\}}\,.$

Definition

A matrix ${}M\in \operatorname {GL} _{n}\!{\left(\mathbb {C} \right)}$ fulfilling

${}{{\overline {M}}^{\text{tr}}}M=E_{n}\,$

is called a unitary matrix. The set of all unitary matrices is called unitary group; it is denoted by

${}\operatorname {U} _{n}\!={\left\{M\in \operatorname {GL} _{n}\!{\left(\mathbb {C} \right)}\mid {{\overline {M}}^{\text{tr}}}M=E_{n}\right\}}\,.$