Geometrically, the effect of the belief adjustment may be represented
by the eigenstructure of a certain linear operator 
defined on [B]. This operator is defined to be
where 
, 
are the orthogonal projections from [D] to [B], and from
[B] to [D], respectively.
is a bounded self-adjoint operator, as 
are adjoint transforms, namely
because both sides of the above equation are equal to (X,Y).
The operator is termed the resolution transform for B
induced by D, as it represents the variance resolved for each X by D as
as
We may also evaluate the transform
where I is the identity operator on [B]. We term 
the variance
transform for B induced by D, as adjusted covariance is represented by
the relation, for each X and Y in 
, that
or equivalently, in terms of the inner products over [B], as
, are self-adjoint operators, of norm at most one. They have
common eigenvectors, 
, with eigenvalues 
,
where 
.
From equation 47, we may deduce that, provided has a discrete spectrum, each canonical direction,
, of the adjustment of B by D, is an eigenvector of , with
eigenvalue 
, and conversely each eigenvector of is a canonical
direction of the adjustment. Thus the eigenstructure of summarises the
effects of the adjustment over the whole structure [B]. In particular, the
resolved uncertainty may be written as
