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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def matrix(self, full=False, keeppads=True):
 """Return matrix form of the integral transform. Parameters full : bool, optional when False return two vector factors and convolution matrix separately, otherwise return full transformation matrix keeppads : bool, optional whether to keep the padding in the output Returns ------- If full is False, output separately a : (1, N) or (1, Nin) ndarray "After" factor, `_yfac_` or `yfac` b : (N,) or (Nin,) ndarray "Before" factor, `_xfac_` or `xfac` C : (N, N) or (Nin, Nin) ndarray Convolution matrix, circulant Otherwise, output the full matrix, combining `a`, `b`, and `C` M : (N, N) or (Nin, Nin) ndarray Full transformation matrix, `M = a * C * b` Notes ----- `M`, `a`, `b`, and `C` are padded by default. Pads can be discarded if `keeppads` is set to False. This is not meant for evaluation with matrix multiplication but in case one is interested in the tranformation itself. When `N` is even and `lowring` is False, :math:`C C^{-1}` and :math:`M M^{-1}` can deviate from the identity matrix because the imaginary part of the Nyquist modes are dropped. The convolution matrix is a circulant matrix, with its first row and first column being the Fourier transform of :math:`u_m`. Indeed :math:`u_m` are the eigenvalues of the convolution matrix, that are diagonalized by the DFT matrix. Thus :math:`1/u_m` are the eigenvalues of the inverse convolution matrix. """


        v = np.fft.hfft(self._u, n=self.N) / self.N
        idx = sum(np.ogrid[0:self.N, -self.N:0])
        C = v[idx]  # follow scipy.linalg.{circulant,toeplitz,hankel}

        if keeppads:
            a = self._yfac_.copy()
            b = self._xfac_.copy()
        else:
            a = self.yfac.copy()
            b = self.xfac.copy()
            C = self._unpad(C, 0, True)
            C = self._unpad(C, 1, False)
        a = a.reshape(-1, 1)

        if not full:
            return a, b, C
        else:
            return a * C * b