Exercise 56 in chapter 1 of Lang's algebra

[stupiduser]

Hi,

I have been puzzled by this exercise for some time. I won't repeat it here as the hint refers to other exercises, so I would be copying a whole page of the book.

My question is as follows. After having defined the sets S_i and picked a complex number s not belonging to any of the S_i it is trivial to use part (b) of exercise 55 to show that for some large k we have M_i^k(s)\in S_i for all i.

Now the problem is that we still need to prove that M_i^{nk}(s)\in S_i for all n\neq 0 i.e. all the exponents, but I don't see how this can be done for negative exponents.

Further, I have no idea how to prove that there exists such a k that M_i^k(S_j)\subset S_i for all j\neq i and further for all the exponents of M_i^k. It doesn't even seem plausible that this could happen as any element close to w'_i is (for a large enough k) mapped into U_i i.e. close to w_i by part (b) of the previous exercise. Thus, it would seem that choosing some z\in U_i' closer and closer to w_i' would require k to grove without bound...

I'm just wondering whether anyone has ever done this exercise or has some tips on how to approach it?

Thanks.

 

[stupiduser]

Nevermind the first part. I just realized that the inverse of a matrix has the same fixed points, so we get the required condition for the element s for negative exponents. This doesn't help with the second problem relating the sets though.