Page 202 - Linear Algebra Done Right

Exercises
Suppose N ∈L(V) is nilpotent. Prove that the minimal poly-
29.
m+1
, where m is the length of the longest con-
nomial of N is z
secutive string of 1’s that appears on the line directly above the 191
diagonal in the matrix of N with respect to any Jordan basis for N.
30. Suppose V is a complex vector space and T ∈L(V). Prove that
there does not exist a direct sum decomposition of V into two
proper subspaces invariant under T if and only if the minimal
polynomial of T is of the form (z − λ) dim V for some λ ∈ C.
31. Suppose T ∈L(V) and (v 1 ,...,v n ) is a basis of V that is a Jordan
basis for T. Describe the matrix of T with respect to the basis
(v n ,...,v 1 ) obtained by reversing the order of the v’s.