Nipotent Operators

Question 1

When is linear operator phi nilpotent

Answer 1

Linear operator phi is nilpotent when there is a basis B of V w.r.t which phi has a strictly upper triangular matrix (0s in diagonals)

Question 2

What is am(L) for a linear operator phi

Answer 2

Am(L) = dim(Gphi(L)) for linear operator phi, since Ephi(L) <= Gphi(L)) explains gm(L) <= am(L)

Question 3

Set phi(i) = phi|Gphi(Li) (restriction), then what is Mphi(i) and Mphi

Answer 3

Mphi(I) = (x - Li)^si some is <= dim(Gohi(Li))
Mphi = I = 1 to k PI Mphi(I) = u = 1 to k (x - Li)^si

Question 4

Given Mphi = PI I = 1 to k (x - Li)^si, what is Gphi(Li)

Answer 4

Given Mphi = PI I = 1 to k (x - Li)^si, Gphi(Li) = ker(phi - Li(idV))^si
Si <= a.m(Li)