I'm having difficulty trying to prove this theorem for my REU research.
I won't go into the full details of the thing I'm trying to prove as it's quite complicated. However, I am trying to solve the inverse of the following n x n matrix:
Yeah, so I am assuming the above won't come out correctly.
1 0 0 0 0 ...
-s_1 1 . . . . . . .
. . 1 . . . . . .
. 0 0 . . . . .
. 0 0 -S_n (1-S_n)
Any way, I want to find the inverse, and therefore the easiest thing to do would be to augment it with the identity n x n matrix.
What the above is supposed to be is (I - T), and thus we have 1's along the diagonal where the last term is (1 - S_n) and -s_1, -s_2, ..., -s_n along the sub-diagonal.
The reason for doing so is it helps me find R_0, the largest positive eigenvalue later.
As I see it, the general pattern is:
1's along the main diagonal, with s_1, s_1*s_2, s_1*s_2*s_3, ... along the sub-diagonal. The issue comes trying to determine the last elements in the matrix.
And then, perhaps the hardest part, would be trying to give a proof of why this is true. Using inducation, I would assume, would be extremely tedious and messy.