The glider.

The glider has period N=4 and hence the cycle matrix A is the product of four transition matrices Tj . It can be shown (by means of a suitable Maple worksheet one can determine the cycle matrix for any reasonably sized initial pattern) that the cycle matrix in the glider case is
A   =         where the order between the cells is defined by    
The cell order is chosen in such a way that the form of the matrix conforms with the canonic form which in the case of a spaceship is :

The four blocks (P, 0, H and S) are indicated in A above.

It is evident from the cycle matrix that the 5th cell is a secondary rotor cell (The zeroes in the 5th column above the diagonal entry imply that cell 5 is not member of a communicating set of cells.) This fact also follows from the properties of the limit matrix B which is determined as follows:
An eigenvector b of PT corresponding to the eigenvalue 1 is easily computed,( e.g. in a Maple worksheet) . In this case we get b=(1,1,3,6)/11
where the component sum is set to 1. Hence b is the generic row of P*.
To get H* we look first at S and (IS - S)-1, which are 1x1 matrices i.e. real numbers, and we get S = 1/81 and (IS - S)-1 = 81/80 respectively.

Hence H* = (IS - S)-1(HP*) =
= (81/80){(2/9 , 1/27 , 1/9 , 50/81) } =
= (1 , 1 , 3 , 6)/11     ( note that the row sum of H is 80/81).

This means that the limit matrix for the glider is:
B   =      =   

Note that the jth column of B indicates the various asymptotic colour indices when the jth cell of the initial pattern is black. The fact that all entries of each column is constant (i.e. B is a -matrix) means that all cells of the glider asymptotically get the same gray shade. The fact that this constant varies between the columns means that different black initial cells create different asymptotic colours. In the glider case we see that the 4th cell creates the darkest cells and hence has the greatest genetic impact.
The zero 5th column shows that the 5th cell is a secondary rotor cell since it has zero genetic impact asymptotically.

The toad

If the cells of the toad are ordered according to the diagram below

the cycle matrix is:
A   =   

This matrix has the general form    

since there are no primary rotor cells.
As before the blocks I,0,G and S are indicated in the A matrix.
To compute B we have to determine G* = (IS - S)-1 G,
We get (IS - S)-1 =     and hence

G*   =      =   .
Finally we get the full limit matrix B   =      =   

Here the two first columns show the impact of the initial black stator cells in positions 1 and 2 on the four secondary rotor cells 3 - 6. The fact that these rotor cells are secondary is reflected by the zeroes in columns 3 - 6 which indicate that the corresponding cells have zero genetic impact.