2011-04-18

(3) Max determinant problem, Appendix

This is just a side talk and you can skip this. It is a detail about relationship between Fredholm equation of the second kind and the max determinant problem. The basic idea is as following.

  • The discrete form of Fredholm equation of the second kind is a matrix form (a linear system).
  • To get the limit of the discrete form, the dimension of matrix n goes to infinity.
  • The solution of linear system is obtained by a variant of Cramer's rule. This needs the determinant.
  • The absolute value of the determinant relates with the system's convergence. Therefore, the max determinant problem was interesting.

Fredholm equation of the second kind is as following.
Assume this equation as a discrete problem, the range a,b is n-subdivided, then
Set λ (b-a)/n = h, the coefficient of this equation becomes following.
When we let the subdivision number n to infinity, the dimension of matrix becomes infinite. When we solve this linear system by Cramer's rule, we need determinant. Please note that Fredholm didn't use Cramer's rule directory. We need a bit more details (Fredholm minor, Fredholm determinant), however, I just want to know what was the motivation of the max determinant problem. (Also I don't understand whole the Fredholm equation discussion.)

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