Linear preservers of balanced nonsingular inertia classes

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Let V be one of the following four real vector spaces: J n , the n × n real symmetric matrices; H n , the n × n complex hermitian matrices; M(n, R) , the n × n real matrices, and M(n, C) , the n × n complex matrices. Suppose T is an R -linear map on V preserving the invertible matrices in the case V = M(n, R) or M(n, C) or preserving the nonsingular balanced inertia class ( n even) in the case V = J n or H n . If n > 2 and n ≠ 4 or 8 when V = M(n, R) , we show that T must be invertible and specify the exact form of T.