Linear Algebra

386 Chapter Five. Similarity̌ 2.26 Show that the only eigenvalue of a nilpotent matrix is zero.2.27 Is there a nilpotent transformation of index three on a two-dimensional space?2.28 In the proof of Theorem 2.14, why isn’t the proof’s base case that the index ofnilpotency is zero?̌ 2.29 Let t: V → V be a linear transformation and suppose ⃗v ∈ V is such thatt k (⃗v) = ⃗0 but t k−1 (⃗v) ≠ ⃗0. Consider the t-string 〈⃗v, t(⃗v), . . . , t k−1 (⃗v)〉.(a) Prove that t is a transformation on the span of the set of vectors in the string,that is, prove that t restricted to the span has a range that is a subset of thespan. We say that the span is a t-invariant subspace.(b) Prove that the restriction is nilpotent.(c) Prove that the t-string is linearly independent and so is a basis for its span.(d) Represent the restriction map with respect to the t-string basis.2.30 Finish the proof of Theorem 2.14.2.31 Show that the terms ‘nilpotent transformation’ and ‘nilpotent matrix’, asgiven in Definition 2.7, fit with each other: a map is nilpotent if and only if it isrepresented by a nilpotent matrix. (Is it that a transformation is nilpotent if anonly if there is a basis such that the map’s representation with respect to that basisis a nilpotent matrix, or that any representation is a nilpotent matrix?)2.32 Let T be nilpotent of index four. How big can the range space of T 3 be?2.33 Recall that similar matrices have the same eigenvalues. Show that the conversedoes not hold.2.34 Lemma 2.1 shows that any for any linear transformation t: V → V the restrictiont: R ∞ (t) → R ∞ (t) is one-to-one. Show that it is also onto, so it is an automorphism.Must it be the identity map?2.35 Prove that a nilpotent matrix is similar to one that is all zeros except for blocksof super-diagonal ones.̌ 2.36 Prove that if a transformation has the same range space as null space. thenthe dimension of its domain is even.2.37 Prove that if two nilpotent matrices commute then their product and sum arealso nilpotent.2.38 Consider the transformation of M n×n given by t S (T) = ST − TS where S is ann×n matrix. Prove that if S is nilpotent then so is t S .2.39 Show that if N is nilpotent then I − N is invertible. Is that ‘only if’ also?