In reply to Ryan McNeive.

Mostly it is a curiosity. I don't think it is important to matrices in the same way that symmetry, positive definiteness, or eigenvalues are.

Saddle points are a particular arrangement among random values. Suppose A[1,1] is the saddle point, then it must be the case that A[1,1] < A[1,j] for each j and A[1,1] > A[i,1] for each i. The rarity of a saddle point is a way of saying that those n+p-2 inequalities are unlikely to happen when the values are selected at random from any distribution. In this way, a saddle point in a matrix can be thought of as a particular 2D arrangement within the set of all possible 2D arrangements. A 1D analogy would be investigating the probability of permutations such as [1 3 5 4 2] that increase monotonically until some value (5) and then decrease monotonically after that value. Is studying such an arrangement "mathematically significant"? Probably not, but it is interesting nontheless.