“Main” diagonal of a matrix

Hello!

I'm in search of some (possibly statistical) measure for matrices. I want to classify a square matrix as having the largest numbers running along the main diagonal or along the anitdiagonal. It isn't necessarily on the diagonals, the numbers can be some rows to the left or right, so a simple trace wouldn't work. I want to incorporate the entire matrix in the calculation.

To see how I came to this problem: the matrix A is a transition matrix, for example in a Markov process (I have a CS background). The probability of going from state i to state j is given by $A_{i,j}$. Each row sums up to 1. The states are ordered though, and I would like to know if a certain matrix makes small or no jumps (largest probabilities along main diagonal) or if huge jumps are present (largest probabilities along antidiagonal).

I don't think this problem is very hard, but I'm at loss for the terminology. Thanks!

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The following very coarse invariant is perhaps useful:

The argument of the sum $\sum_{s,t}a_{s,t}e^{i(s-t)\pi/n}$ (where $n$ is the size of the matrix and where the sum is over all entries) should be related to a typical jump. It is close to $0$ if the matrix is "concentrated" near the diagonal and close to $\pi$ if the matrix is "concentrated" near the antidiagonal. The modulus of the above sum indicates somehow how much the matrix is concentrated.

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On thinking it through, this seems pretty much what I search. Thanks! –  Verhoevenv May 27 '10 at 21:11
Note that the question isn't an either-or between the diagonal and the anti-diagonal. There are many things that a transversal can do other than be close to the diagonal or anti-diagonal. Even so, you can look at how close the transversal is to one or the other extreme. For instance, you can look at its inversion number. The diagonal has the smallest inversion number, 0, while the antidiagonal has the unique largest inversion number, $n(n-1)/2$.