balance

[Peter Cameron]

Just a brief note to clarify what I didn't make clear before.

JP said:

> > A matrix is t-balanced if it has constant diagonal and constant off-
> > diagonal. .... each type of t-balance is what is already called balance of 
> >that type.
> 
> This one doesn't work out so nicely in the unequal replication case for the
> matrix R^{-1/2}NR^{-1/2}. One of the eigenvectors of this matrix, with
> eigenvalue 0, is the diagonal of R^{-1/2}. So an efficiency balanced design
> (v-1 identical, nonzero efficiency factors) need not have the property of
> constant diagonal and constant off-diagonal. The efficiency factors are also
> eigenvalues of R^{-1}N, for which (like N) the "structural 0" eigenvalue now
> goes with the constant vector, but R^{-1}N need not be symmetric.

I wasn't claiming any implications between the three types of balance
(pairwise, variance, and efficienccy balance); only that the concepts
of pairwise/variance/efficiency partial balance with the trivial
association scheme reduce precisely to the previously-defined concepts
of pairwise/variance/efficiency balance (if I have understood the
definitions correctly). 

Peter.