A Method for Calculating the Gini coefficient

Determine the determinant $ \ det A = \ det \ left (\ frac {1} {A _ {I} + B _ {J} \ right) _ {n \ times N }$ $ is the determining factor of the kernel. We will calculate it:

Because $ \ det \ left (\ frac {1} {A _ {I} + B _ {J} \ right) _ {n \ times N }=\ frac {1} {\ prod \ limits _ {1 \ Leq I, J \ Leq n} (A _ {I} + B _ {J})} \ det (C _ {IJ}) _ {n \ times n} $

If $ A _ {1}, \ cdots, A _ {n}, B _ {1}, \ cdots, and B _ {n} $ are treated as variables, every $ C _ {IJ} $ Is A Multivariate Polynomial of $ n-1 $ times. Therefore, $ \ det (C _ {IJ }) the result of _ {n \ times n} $ must be a Multivariate Polynomial of $ n (n-1) $ times $ f (a _ {1}, \ cdots, A _ {n}, B _ {1}, \ cdots, B _ {n}) $ (simplified as $ F $ ).

If we treat $ A _ {1} $ as a variable and the rest as a constant, it is clear that if $ A _ {k} = A _ {1} $ exists, $ F = \ det (C _ {IJ}) _ {n \ times n} = 0 $

Therefore, $ F $ has a factor $ (A _ {1}-A _ {k}), k = 2, 3, \ cdots, N $; similarly, we can see that $ F $ contains a factor $ G = \ prod _ {1 \ Leq I <J \ Leq n} (A _ {I}-A _ {J }) (B _ {I}-B _ {J}) $

Note that $ {\ RM deg} G = 2 \ binom {2} {n }={ \ RM deg} f $

Therefore, the difference between $ F $ and $ G $ is only one non-zero constant coefficient. that is, $ \ det A =\frac {c \ prod \ limits _ {1 \ Leq I <J \ Leq n} (A _ {I}-A _ {J }) (B _ {I}-B _ {J})} {\ prod \ limits _ {1 \ Leq I, J \ Leq n} (A _ {I} + B _ {J}) $ let's find this coefficient $ C $, now $ A _ {I }=\ frac {1} {2} + ix, B _ {J }=\ frac {1} {2}-JX $, then $ \ det A = \ det \ left (\ frac {1} {1 + (I-j) x} \ right) _ {n \ times n} $

Note that the above formula results are continuous for the sufficiently large $ x $, so that we can know $ x \ To \ infty $ \ det A = 1 $

At this time, \ begin {Align *} \ lim _ {x \ To \ infty} \ frac {\ prod \ limits _ {1 \ Leq I <J \ Leq n} (_ {I}-A _ {J }) (B _ {I}-B _ {J})} {\ prod \ limits _ {1 \ Leq I, J \ Leq n} (A _ {I} + B _ {J })} & =\ LiM _ {x \ To \ infty} \ frac {\ prod \ limits _ {1 \ Leq I <J \ Leq n} (I-j) X (J-I) x} {\ prod \ limits _ {1 \ Leq I, j \ Leq n} (1 + (I-j) x )} \\&=\ frac {-\ prod \ limits _ {1 \ Leq I <J \ Leq n} (I-j) ^ 2} {\ prod \ limits _ {1 \ Leq I, j \ Leq N, I \ neq j} (I-j )} = 1 \ rightarrow C & = 1 \ end {Align *}

In summary, $ \ det \ left (\ frac {1} {A _ {I} + B _ {J} \ right) _ {n \ times N }=\ frac {\ prod \ limits _ {1 \ Leq I <J \ Leq n} (A _ {I}-A _ {J }) (B _ {I}-B _ {J})} {\ prod \ limits _ {1 \ Leq I, J \ Leq n} (A _ {I} + B _ {J}) }$ $

A Method for Calculating the Gini coefficient