Let be the set of all complex matrices which are hermitian, that is, , where .
(a) Show that is a real 4-dimensional vector space. Consider the real symmetric bilinear form on this space defined by
Prove that and , where denotes the identity matrix.
(b) Consider the three matrices
Prove that the basis of diagonalizes . Hence or otherwise find the rank and signature of .
(c) Let be the set of all complex matrices which satisfy . Show that is a real 4-dimensional vector space. Given , put
Show that takes values in and is a linear isomorphism between and .
(d) Define a real symmetric bilinear form on by setting , . Show that for all . Find the rank and signature of the symmetric bilinear form defined on .