Speaker
Description
This talk presents spectral properties of monotone stochastic matrices which are characterised by the fact that each row stochastically dominates the preceding one, and which arise in contexts such as intergenerational mobility, equal-input models, and credit-rating systems.
In analogy with the stochastic matrices, for the monotone stochastic matrices both the individual eigenvalues as the spectrum as a whole are examined. Individually, the eigenvalue region for all $n \times n$ monotone matrices with $1 \leq n \leq 3$ is completely determined, and realising matrices are provided. Collectively, the set of possible pairs of non-trivial eigenvalues arising from $3 \times 3$ monotone matrices is characterised, accompanied by realising matrices. In both perspectives, the resulting regions are substantially smaller than those for general stochastic matrices. Finally, a reduction theorem is presented, stating that, for $n \geq 4$, the eigenvalue region of $n \times n$ monotone matrices is contained within that of $(n-1) \times (n-1)$ stochastic matrices.