Odd-dimensional special orthogonal transformation has a fixed vector
= Odd-dimensional special orthogonal transformation has a fixed vector
Every real <orthogonal transformation> of odd dimension with <determinant> one has <eigenvalue> $+1$. Nonreal <eigenvalues> come in conjugate pairs with product one, while real <eigenvalues> are $+1$ or $-1$. There are an odd number of real <eigenvalues> and an even number of negative ones, leaving at least one positive <eigenvalue>. The corresponding <eigenvector> is fixed.