Title: On the validity of parametric block correlation matrices with constant within and between group correlations,
Abstract: We consider the set Bp of parametric block correlation matrices with p blocks
of various (and possibly different) sizes, whose diagonal blocks are compound
symmetry (CS) correlation matrices and off-diagonal blocks are constant
matrices. Such matrices appear in probabilistic models on categorical data,
when the levels are partitioned in p groups, assuming a constant correlation
within a group and a constant correlation for each pair of groups. We obtain
two necessary and sufficient conditions for positive definiteness of elements
of Bp. Firstly we consider the block average map $\phi$, consisting in
replacing a block by its mean value. We prove that for any A $\in$ Bp , A is
positive definite if and only if $\phi$(A) is positive definite. Hence it is
equivalent to check the validity of the covariance matrix of group means, which
only depends on the number of groups and not on their sizes. This theorem can
be extended to a wider set of block matrices. Secondly, we consider the subset
of Bp for which the between group correlation is the same for all pairs of
groups. Positive definiteness then comes down to find the positive definite
interval of a matrix pencil on Sp. We obtain a simple characterization by
localizing the roots of the determinant with within group correlation values.