In reply to Rick Wicklin.

I understand saddle points in search algorithms a little via local minimums and maximums. We search for these points as solutions in some work. In that context would the saddle points not be very significant in some domains such as optimizations? Say we are optimizing something to be minimum in one dimension but maximum in some other dimension then finding the saddle points is finding our solution.
Now back to matrices. Are matrices not useful for mapping Markov chains? Now I am in the unknown for me a bit but speculating that some how saddle points would be useful in matrices representing Markov chains.