<P> degrees of freedom, and so does O (n). </P> <P> Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order n! symmetric group S. By the same kind of argument, S is a subgroup of S . The even permutations produce the subgroup of permutation matrices of determinant + 1, the order n! / 2 alternating group . </P> <P> More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two - dimensional subspaces . That is, if Q is special orthogonal then one can always find an orthogonal matrix P, a (rotational) change of basis, that brings Q into block diagonal form: </P> <Dl> <Dd> P T Q P = (R 1 ⋱ R k) (n even), P T Q P = (R 1 ⋱ R k 1) (n odd). (\ displaystyle P ^ (\ mathrm (T)) QP = (\ begin (bmatrix) R_ (1) && \ \ & \ ddots & \ \ &&R_ (k) \ end (bmatrix)) \ (n (\ text (even))), \ P ^ (\ mathrm (T)) QP = (\ begin (bmatrix) R_ (1) &&& \ \ & \ ddots && \ \ &&R_ (k) & \ \ &&&1 \ end (bmatrix)) \ (n (\ text (odd))).) </Dd> </Dl>

If p q are two orthonormal matrices then so is p q