Before revision: We recall that if A is an invertible and symmetric real m\times m matrix, then it is diagonalisable. Therefore, if we denote by \mathcal{E^{+}(A)\subset R^m (respectively E^{-}(A)\subset R^m) to be the vector subspace generated by eigenvectors with positive eigenvalues of A (correspondingly the vector subspace generated by eigenvectors with negative eigenvalues of A), then we have an orthogonal decomposition R^m=E^{+}(A)\oplus E^{-}(A).
After revision: We ^{+}(A)\subset R^m (respectively E^{-}(A)\subset R^m) to be the vector subspace generated by eigenvectors with positive eigenvalues of A (correspondingly the vector subspace generated by eigenvectors with negative eigenvalues of A), then we have an orthogonal decomposition R^m=E^{+}(A)\oplus E^{-}(A).