Exercise 3.2.10: Let A ∈ F^{n×n} and consider the linear map A ∈ L(F^n,F^n) determined by the n×n matrix
0 0 0 0 ... 0 0
0
0 0 0 ...
0
L0 0 0
.. ..
.. ... ... ... ... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ... 0 0 1 0 -.. 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
A
.
.-
0 0 0
0
0
0 0 0
0 0
1 0
0
0
We suppose the lower right block is a k×k matrix and the upper left block, therefore, is an (n-k)×(n-k) matrix. Answer the following questions.
a) What are the eigenvalues of A?
(b) For each of the eigenvalues of A, determine its algebraic multiplicity.
(c) For each of the eigenvalues of A, determine its eigenspace.
(d) For each of the eigenvalues of A, determine its geometric multiplicity.
(e) For each of the eigenvalues of A, determine its generalized eigenspace.
(f) For each of the eigenvalues f of A, determine the smallest m ∈ ℤ₀ for which WeA = ker(A - fI_m).