The

column rank of A is the generator dimension for the column space of A.

However, there are two distinct limitations in this algorithm: 1) The successful decoding ratio is entirely dependent on the rank of the random matrix (the coefficient matrix of all the collected encoded packets).Only when the random matrix is full

column rank, can user decode data successfully and recover all the source packets at one time.

is of full

column rank. Due to [[A.bar].sub.22] being nonsingular,

A matrix m has full

column rank if mm is invertible.

It is shown that C is full

column rank because the system is observable while B is full

column rank because the system is controllable.

where x(t) [member of] [R.sup.n] is state vector, u(t) [member of] [R.sup.m] is control input, and y(t) [member of] [R.sup.l] is controlled output, E [member of] [R.sup.nxn] is singular matrix with rank E = [n.sub.1] [less than or equal to] n, and B [member of] [R.sup.nxm], C [member of] [R.sup.lxn] are full

column rank and full row rank, respectively.

We denote [j.sub.k] = min{j | j [member of] I([x.sup.k])} and use the following pivoting operation to generate the index set [[??].sup.k] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has full

column rank, so vectors {[g.sub.j]([x.sup.k]), j [member of] [[??].sub.k]} are linearly independent.

The system (1) is said to be left invertible if and only if its transfer function matrix (13) has full

column rank.

The maximal number of right independent columns of A will be called right

column rank of A.

A system is observable if [[GAMMA].sub.0] has full

column rank n and, in turn, a system is controllable if [[GAMMA].sub.c] has full row rank.

P is called column reduced, if there exists a permutation matrix T, such that P = (0, [P.sub.1])T, where [[Gamma].sub.c]([P.sub.1]) has full

column rank.