(7), [V.sup.N] spans the
nullspace of H and is the orthogonal complement of [V.sup.S].
where [J.sup.+] is the pseudoinverse of J, [P.sup.[perpendicular to]] (J) is the projection operator which projects arbitrary joint rates into the
nullspace of the end-effector's Cartesian coordinates, and [xi] is an arbitrary joint rate vector.
The normalized eigen vectors corresponding to the zero eigen values of [S.sub.NC.sup.i[phi]] form a basis of the
nullspace of [S.sub.NC.sup.i[phi]].
The dimension of the eigenvalue [lambda] =1 is n + [n.sub.0], where [n.sub.0] is the dimension of the nontrivial
nullspace of [B.sub.1] + [B.sub.2].
Given a(n) and b(n) in (14), the ICI reconstructing vector [v.sup.H.sub.1(n)] is determined by the left-hand
nullspace of the matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
and we are only looking for eigenvectors in the
nullspace of [U.sup.*.sub.2]M.
We note at this point that this matrix is in general singular, as it is well known that the vector of ones is a member of the
nullspace of [B.sup.T] (see [6, Chapter 5] for instance)--the matrix in (3.1) therefore has two zero eigenvalues (one corresponding to each appearance of [B.sup.T]).
However, since [[bar.H.]sub.l] [member of] [R.sup.4 x 4] is a square matrix with full rank, its
nullspace, which is also the intersection subspace of span([[??].sub.i]) and span([[??].sub.j]).
where F [member of] [R.sup.nxk] is a matrix the columns of which form an orthonormal basis of the
nullspace of L; cf.
A is a matrix, [a.sup.m] is a mth column vector of A, [A.sup.H] denotes the conjugate transpose of A, and [parallel]A[[parallel].sub.F] is its Frobenius norm; tr(A) is its trace, span (A) is its column space, null(A) returns
nullspace vectors of column space of A, eig(A) is any eigenvector of A, O(A) consists of the orthonormal basis vectors that span the column space of A; [I.sub.M] is the M x M identity matrix; [C.sup.M] is the M-dimensional complex vector space.
By using an orthogonal projection P whose
nullspace is Z the Krylov space solver is then applied only to the orthogonal complement [Z.sup.[perpendicular to]] by restricting the operator A accordingly.
where [[phi].sub.ij] [member of] [0,1] are uniformly distributed random numbers, [OMEGA] = {1, ..., n}x{l, ..., m}, [summation] = {1, ..., l} x {1, ..., m} and l >> n, such that [bar.K] has a large
nullspace. Furthermore, we consider f instead of g, with f being corrupted by Poisson noise.
