Test whether matrixes are equal

[fer2000]

Hello everybody!

I have a large set of items and, two-by-two, there exists some kind of attraction/repulsion that I have measured.
Under different circumstances this attraction/repulsion changes.
So, what kind of statistical/mathematical procedure can I use to test whether the changes (for the whole set) are “large/small”?
If I want to test whether in a subset of them (lets say the first ten items) the attraction is larger than in another subset (lets say the last ten items), how can I do it?.

Thanks in advance

 

[xag]

1) If A is your set of n items, For (a1, a2) in AxA and a date t in R, att(a1, a2, t) in R gives the attraction (>0) or repulsion (<0) between a1 and a2 at date t.

Build a vector V in AxA that spans all possible (a1, a2) with a1 and a2 both in A. V = (v1, ..., vn*n).

Now for each t in R, you can define Wt in R^(n*n) so that Wt = (w1, ..., wn*n) and for i in [1; n*n], wi = att(ax, ay, t) with vi = (ax, ay).

A metric to reveal the attraction changes can be a distance on R^(n*n).It will tell you how far apart are two elements Wt1 and Wt2: how big was the interaction change.

2) For the second part you fix t.
Instead of a vector V you build a matrix M with an element of A indexing each column and an element of A indexing each row, and the distance between these elements in the cases.
M is in Rn,n.
You take the PCA of M and you make sure the eigenvalues are > 0. The eigenvalues you end up with represent principal attractions in the whole set. For each principal attraction, the corresponding eigenvector is a linear combination of some items, and the coefficient for each item in this linear combination tells you to what extent this item is concerned by this attraction, and in which direction (if the coef is <0 the item is repulsed).

So the items present in the eigenvalue corresponding to the largest eigenvector are those who interact the most, and among them the ones with the biggest coefficient are the ones that contribute the most to this interaction.

Note well: this is merely intuitive and I didn't prove anything..