In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e.

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V ↦ W between two modules (includingvector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. 

In mathematics, the orthogonal group of dimension n, denoted O(n), is the group of distance-preserving transformations of aEuclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. 

Equivalently, it is the group of n×n orthogonal matrices of a given dimension, where the group operation is given by matrix multiplication, and an orthogonal matrix is a real matrix whose inverse equals its transpose.

The determinant of an orthogonal matrix being either 1 or −1, an important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal matrices of determinant 1.