The representations of the de Bruijn matrix as sums and products need very little change to apply to the probabilistic versions as well. The factorizations give immediate formulas for determinants; when they do not vanish, the block diagonal form of of one of the factors, together with the fact that the other is a permutation matrix quickly reveals a form for the inverse matrix. Taking as an example the matrix of the last section, we have

The Gerschgorin limit of this matrix requires its maximum eigenvalue to
be less than the greatest reciprocal determinant or
(all with respect to absolute values). Generalizing, this
means that * every* eigenvalue of must be greater than
the lesser of these determinants. Of course, if one of them is zero, we
know that the lower bound is reached and is singular.

In the other direction, unless some or is zero, every matrix element of will be strictly positive, although in the probabilistic case no two of them have to be equal. This ensures that the maximum eigenvalue, , will be unique. Conversely, if some probability or coprobability vanishes, the possibility of degeneracy exists (but is not obligatory).

E-mail:mcintosh@servidor.unam.mx