It is worth reviewing some of the tricks which will be used in the following development.
Most of these are used so repetitiously that it would extremely laborious and aesthetically
abhorrent to document each individual application. Firstly, the properties of the Outer Product
will be used almost continuously (see section 2.1.1). Secondly, we will be attempting to simplify things by decomposing
mathematical objects into components, some of which will go to zero. Generally, things will go
to zero because of their independance, and the fact that the Outer Product of independant RV's is
the Zero Matrix. There are two general ways we will see independence arise: the first, which I call
Structural Orthogonality, is the standard independence of completely unrelated variables. For
instance, we are assuming the plant and measurement noise to be structurally independent. Other times,
we will claim orthogonality between objects which are clearly related. This is possible if the objects
are made independent by a difference in time index. For instance, the 
observation is independent
of the the 
measurement noise RV. I call this Time Orthogonality. Most of what transpires below
can be explained in terms of the properties reviewed here. So let us begin this final campaign,
strike out for that tauntingly closer goal toward which we strive. We're almost there.
We begin by focusing our attention on 
. This sequence, while unavailable
for direct observation, contains all the information about behavior of the system in the
sense that it yields a deterministic knowledge of the systems path through state space. We
will base our model of the systems behavior on the statistical behavior of this sequence.
To begin, we will express the covariance matrices used in terms of the covariance of
, denoted 
.
This decomposition of the vector allows us to analyse its covariance as follows:
Now, in order to similarly express the second covariance matrix needed in terms
of 
, we again begin with a familiar decomposition:
