Starting with the evolution and de Bruijn matrices, and continuing with their probabilistic versions, we find that we are dealing with a specialized class of matrices all of whose elements are positive---or more accurately, non-negative. The first two of these classes of matrices is more restricted, inasmuch as their elements must be integers, but it is not especially easy to obtain much advantage from that particular characteristic. However, positive matrices---including those with integer elements---enjoy two properties which can be exploited to considerable advantage. Avoiding some limiting cases arising from unfavorable groupings of zero matrix elements, these properties are:

- there is a unique maximum eigenvalue, whose value is bracketed by
the row sums of the matrix (as well as the column sums), and
- whose eigenvector can be normalized so that all its components
are strictly positive. That is, all have the same sign and none are
zero.

- Gerschgorin's disks
- Eigenvalues on the boundary
- Minimax principle
- Largest eigenvalue
- Second largest eigenvalue
- Averaging and convergence
- Non-negative matrices

E-mail:mcintosh@servidor.unam.mx