The generic form of a matrix with unit column sum is

whose eigenvalues are

whose matrix of left eigenvectors is

and whose matrix of right eigenvectors is

The eigenvalue 1, whose eigenvector has components proportional to the off diagonal elements of the matrix, represents an equilibrium probability. The second eigenvalue lies in the range with an eigenvector suitable for probability differences. Indeed it is the factor by which the disequilibrium decreases in each generation; if it is zero equilibrium is reached in one step, otherwise there will be an exponential approach to equilibrium. Depending on the sign, there may be oscillations about equilibrium, or a uniform approach to equilibrium. However, if the second eigenvalue is as large as one, there will either be a degeneracy by which any probability is in equilibrium, or the probabilities for zeroes and ones will be exchanged with each other and alternate forever after.

A doubly stochastic one-stage de Bruijn diagram would have a probability matrix

with **q = 1 - p**, whose eigenvalues would be ;
being symmetric, the matrix of both left and right eigenvectors would be

The probability matrix of a two stage diagram is a little more complicated.

with and the characteristic polynomial

with two determinants defined by

The determinant of the de Bruijn matrix is the product of two smaller
determinants corresponding to the evident blocks in the de Bruijn
matrix; in this we have a special case of a quite general result. The
**p**'s and **q**'s were defined as they are with the thought of equating
the subscript pairs (1,2) and (3,4) to get a doubly stochastic matrix,
but the best interpretation of the vanishing of the small determinants
is that the biases of the probabilities in their submatrices are equal.

The constant term in the characteristic equation will be zero, giving a single zero root, if either determinant is zero. The coefficient of will also be zero if both are zero, producing a double zero root. In order to get three equal roots and thus open up the possibility of the Jordan normal form, requires some similarity between the two submatrices in addition.

E-mail:mcintosh@servidor.unam.mx