Orthogonal matrix/Orthogonal group/Field/Definition

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

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

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

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