In this paper, the previous definition of the Reverse Jacket matrix (RJM) is revised and generalized. In particular, it is shown that the inverse of the RJM can be obtained easily by a constructive approach similar to that used for the RJM itself. As new results, some useful properties of RJMs, such as commutativity and the Hamiltonian symmetry appearing in half the blocks of a RJM, are shown, and also 1-D fast Reverse Jacket transform (FRJT) is presented. The algorithm of the FRJT is remarkably efficient than that of the center-weighted Hadamard transform (CWHT). The FRJT is extended in terms of the Kronecker products of the Hadamard matrix. The 1-D FRJT is applied to the discrete Fourier transform (DFT) with order 4, and the N-point DFT can be expressed in terms of matrix decomposition by using 4 4 FRJT.