Prove that the eigenvalues of a Hermitian matrix are real and that eigenvectors corresponding to distinct eigenvalues are orthogonal (i.e. ).
Let be a real non-zero antisymmetric matrix. Show that is Hermitian. Hence show that there exists a (complex) eigenvector such , where is imaginary.
Show further that there exist real vectors and and a real number such that
Show also that has a real eigenvector such that .
Let . By considering the action of on and , show that is a rotation matrix.