Let be a real matrix such that , and , where is the transpose of and is the identity.
Show that the set of vectors for which forms a 1-dimensional subspace.
Consider the plane through the origin which is orthogonal to . Show that maps to itself and induces a rotation of by angle , where . Show that is a reflection in if and only if has trace 1 . [You may use the fact that for any invertible matrix B.]
Prove that .