Orthogonal Group

The set of matrices that are orthogonal, or

$$O{O}^T=1$$

forms a group. We can see this by first analyzing the determinant property of orthogonal matrices,

$$\text{det}(O)=\pm 1$$

which can be seen by the orthogonality condition and properties of determinants. An orthogonal matrix whose determinant is $+1$ is called proper and represents rotations, while a matrix whose determinant is $-1$ is called improper and represents mirror operations.