Orthogonal matrix
An orthogonal matrix is a square matrix G whose entries are real numbers and whose transpose is its inverse, i.e.,
- GG'=G'G=In
where G' is the transpose of G. This is the same as saying that the columns of G form an orthonormal basis of Rn. It is an easy exercise, relying on basic facts about inversion of matrices, to show that that is the same as saying that the rows also form an orthonormal basis. The set of all nxn orthogonal matrices forms a group, i.e., it is closed under inversion of matrices and under matrix multiplication. It is a Lie group of dimension n(n+1)/2. The determinant of any orthogonal matrix is 1 or -1. That can be shown as follows:
- 1=det(I)=det(GG')=det(G)det(G')=(det(G))2.
The set of all orthogonal matrices whose determinant is 1 is a subgroup of index 2.