Finite Boundary Value Problems Solved by Green's Matrix.

Abstract

If A=\pmatrix{a_{00}&a_{01}\cr a_{10}&a_{11}&\ddots\cr&\ddots&\ddots} symmetric, thenA^{-1}=\pmatrix{u_0v_0&\cdots\span&u_0v_n\cr&u_1v_1&\cdots&u_1v_n\cr \vdots&\vdots&&\vdots\cru_0v_n&u_1v_n&\cdots&u_nv_n}/W where u and v aredetermined from (Au)_i=0 for i 0,and W=a_{kj}\bigl|{a_{aj}\atop a_{ak}}{a_{bj}\atop a_{bk}}\bigr|.