where [THETA] = [[[THETA].sub.1], [[THETA].sub.2], [[THETA].suub.3]].sup.T] and

antisymmetric matrixFor a 2n x 2n antisymmetric complex matrix A, there is a decomposition A = U[summation][U.sup.T], where U is a unitary matrix and [summation] is a block-diagonal antisymmetric matrix with 2 x 2 blocks:

where * M computes the Hodge dual of a 4 x 4 antisymmetric matrix M; that is, [(*M).sub.ab] = (1/2)[[epsilon].sub.abcd][M.sup.cd].

Because any 6x6 antisymmetric matrix of rank 4 spans a four-dimensional subspace [R.sup.4] [subset] [R.sup.6], the operator (71) in this case can be written in the four-dimensional subspace as

Note that [I.sub.AB] is a 6x6 antisymmetric matrix of rank 6.

where [C.sup.b(l-1).sub.b(l)] is a transformation matrix that transforms vectors from the body frame at cycle I into the body frame at cycle I - 1 and [[phi].sub.l] x is cross-product

antisymmetric matrix composed of [[phi].sub.l] components.

is an

antisymmetric matrix, and [A.sub.[xi]] = diag{[[xi].sub.11], ..., [[xi].sub.mm]}, where [[xi].sub.ii] = [a.sub.ii] - [[alpha].sub.ii] - [[beta].sub.ii], i = 1, ..., m.

The nonzero entries in the

antisymmetric matrix indicate that the principal meridians of the lens surface are not perpendicular.

We note that [D.sub.1] is a real

antisymmetric matrix. Using the spectral differentiation matrix, we obtain the standard Fourier pseudospectral semi-discretization for the nonlinear Schrodiger equation:

For the subsequent analysis, we decompose L into L = [L.sup.+] + [L.sup.-], where symmetric matrix [L.sup.+] = [([l.sup.+.sub.ij]).sub.NxN] and

antisymmetric matrix [L.sup.-] = [([l.sup.-.sub.ij]).sub.NxN] satisfy the zero-row-sum condition with nondiagonal entries

where [[theta].sub.[mu]v] stands for the constant

antisymmetric matrix. However, for the twisted algebra the

antisymmetric matrix [[theta].sub.[mu]v] is no longer constant and depends on coordinates.

where [[theta].sup.ij] = [theta][[epsilon].sup.ij] is an

antisymmetric matrix with real elements and represents the noncommutativity of the space.