The 2X2 identity matrix has every unit vector as an eigenvector, but most symmetric matrices near the Identity matrix have just 2 orthogonal unit vectors as eigenvectors. Can you figure out someway to make its eigenvectors be any unit vectors?

Show that if n is given, there exists a symmetric matrix S, s.t. I+tS has eigenvectors (cos(n),sin(n)) and (sin(n),-cos(n)) for any non-zero t

The 2 X 2 identity matrix has every unit vector as an eigenvector, but most symmetric matrices near the identity have just two orthogonal unit vectors (and their opposites) as eigenvectors. They could in fact be any unit vectors: show that if 0 is given, there exists a symmetric matrix S such that I + 63 has eigenvectors (cos 0, sin 6') and (sin (9, cos 0) for any 6 7E 0