Reduce integer array mat's determinant mag times by linear combination
        of its row vectors, so that the new array after rotation (r_matrix) is
        still an integer array

        Args:
            mat (3 by 3 array): input matrix
            mag (integer): reduce times for the determinant
            r_matrix (3 by 3 array): rotation matrix
        Return:
            the reduced integer array

def reduce_mat(mat, mag, r_matrix):
        """
        Reduce integer array mat's determinant mag times by linear combination
        of its row vectors, so that the new array after rotation (r_matrix) is
        still an integer array

        Args:
            mat (3 by 3 array): input matrix
            mag (integer): reduce times for the determinant
            r_matrix (3 by 3 array): rotation matrix
        Return:
            the reduced integer array
        """
        max_j = abs(int(round(np.linalg.det(mat) / mag)))
        reduced = False
        for h in range(3):
            k = h + 1 if h + 1 < 3 else abs(2 - h)
            l = h + 2 if h + 2 < 3 else abs(1 - h)
            j = np.arange(-max_j, max_j + 1)
            for j1, j2 in itertools.product(j, repeat=2):
                temp = mat[h] + j1 * mat[k] + j2 * mat[l]
                if all([np.round(x, 5).is_integer() for x in list(temp / mag)]):
                    mat_copy = mat.copy()
                    mat_copy[h] = np.array([int(round(ele / mag)) for ele in temp])
                    new_mat = np.dot(mat_copy, np.linalg.inv(r_matrix.T))
                    if all([np.round(x, 5).is_integer() for x in list(np.ravel(new_mat))]):
                        reduced = True
                        mat[h] = np.array([int(round(ele / mag)) for ele in temp])
                        break
            if reduced:
                break

        if not reduced:
            warnings.warn("Matrix reduction not performed, may lead to non-primitive gb cell.")
        return mat