Linear Algebra

Section III. Nilpotence 377
�
−2
(a)
−1
�
45
�
4
2
−22
�
3
(b)
1
�
−19
�
1
3
(c)
�
−3
−3
−3
2
2
2
�
1
1
1
(e) 33 −16 −14
69 −34 −29
� 2.20 Find the canonical form of
⎛
this matrix.
0 1 1 0
⎞
1
⎜0
⎜
⎜0
⎝0
0
0
0
1
0
0
1
0
0
1 ⎟
0⎟
0⎠
0 0 0 0 0
(d)
�
1 1
�
4
3 0 −1
5 2 7
� 2.21 Consider the matrix from Example 2.16.
(a) Use the action of the map on the string basis to give the canonical form.
(b) Find the change of basis matrices that bring the matrix to canonical form.
(c) Use the answer in the prior item to check the answer in the first item.
� 2.22 Each of these matrices is nilpotent.
� �
1/2 −1/2
(a)
(b)
1/2 −1/2
� 0 0 0
0 −1 1
0 −1 1
�
(c)
�
−1 1
�
−1
1 0 1
1 −1 1
Put each in canonical form.
2.23 Describe the effect of left or right multiplication by a matrix that is in the
canonical form for nilpotent matrices.
2.24 Is nilpotence invariant under similarity? That is, must a matrix similar to a
nilpotent matrix also be nilpotent? If so, with the same index?
� 2.25 Show that the only eigenvalue of a nilpotent matrix is zero.
2.26 Is there a nilpotent transformation of index three on a two-dimensional space?
2.27 In the proof of Theorem 2.13, why isn’t the proof’s base case that the index
of nilpotency is zero?
� 2.28 Let t: V → V be a linear transformation and suppose �v ∈ V is such that
t 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 the
span. 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.
2.29 Finish the proof of Theorem 2.13.
2.30 Show that the terms ‘nilpotent transformation’ and ‘nilpotent matrix’, as
given in Definition 2.6, fit with each other: a map is nilpotent if and only if it is
represented by a nilpotent matrix. (Is it that a transformation is nilpotent if an
only if there is a basis such that the map’s representation with respect to that
basis is a nilpotent matrix, or that any representation is a nilpotent matrix?)
2.31 Let T be nilpotent of index four. How big can the rangespace of T 3 be?
2.32 Recall that similar matrices have the same eigenvalues. Show that the converse
does not hold.
2.33 Prove a nilpotent matrix is similar to one that is all zeros except for blocks of
super-diagonal ones.