  The ranks and kernels of generalized Hadamard matrices are studied. It is
proven that any generalized Hadamard matrix $H(q,\lambda)$ over $F_q$, $q>3$,
or $q=3$ and $\gcd(3,\lambda)\not =1$, generates a self-orthogonal code. This
result puts a natural upper bound on the rank of the generalized Hadamard
matrices. Lower and upper bounds are given for the dimension of the kernel of
the corresponding generalized Hadamard codes. For specific ranks and dimensions
of the kernel within these bounds, generalized Hadamard codes are constructed.
