Orthogonal matrix/Orthogonal group/Field/Definition

Orthogonal group

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

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