Euclidean vector space/Proper Isometry/Introduction/Section

Definition

An isometry on a euclidean vector space is called proper if its determinant

is ${}1$.

An isometry that is not proper, that is, its determinant is ${}-1$, is also called an improper isometry.

Definition

Let ${}K$ be a field, and ${}n\in \mathbb {N} _{+}$. An orthogonal ${}n\times n$-matrix ${}M$ fulfilling

${}\det M=1\,$

is called a special orthogonal matrix. The set of all special orthogonal matrices is called special orthogonal group;

it is denoted by ${}\operatorname {SO} _{n}\!{\left(K\right)}$.

Definition

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

${}\det M=1\,$

is called a special unitary matrix. The set of all special unitary matrices is called Special unitary group;

it is denoted by ${}\operatorname {SU} _{n}$.