In reply to Peter Timusk.

Yes, saddle points are important in optimization. A local extremum can be classified as a maximum, a minimum, or "something else", and a saddle point is an important sub-case of the "something else" category. For a continuous objective function, the first derivatives vanish at a local extremum and you use second derivatives to classify the type of extremum.

Regarding Markov transition matrices, if the cell (i,j) is a saddle point then it means that when the system is in "State i", it is least likely to transition to "State j" (minimum of row i). However, every other state has a higher probability of transitioning to "State j" (maximum of column j).