Strings

What is the index of nilpotency of the right-shift operator, here acting on the space of triples of reals?

For each string basis state the index of nilpotency and give the dimension of the range space and null space of each iteration of the nilpotent map.

• $\begin{array}{ccccccc}\vec{\beta}_{1}&\mapsto&\vec{\beta}_{2}&\mapsto&\vec{0}\\ \vec{\beta}_{3}&\mapsto&\vec{\beta}_{4}&\mapsto&\vec{0}\end{array}$

• $\begin{array}{ccccccc}\vec{\beta}_{1}&\mapsto&\vec{\beta}_{2}&\mapsto&\vec{\beta}_{3}&\mapsto&\vec{0}\\ \vec{\beta}_{4}&\mapsto&\vec{0}\\ \vec{\beta}_{5}&\mapsto&\vec{0}\\ \vec{\beta}_{6}&\mapsto&\vec{0}\end{array}$

• $\begin{array}{ccccccccc}\vec{\beta}_{1}&\mapsto&\vec{\beta}_{2}&\mapsto&\vec{\beta}_{3}&\mapsto&\vec{0}\end{array}$

Also give the canonical form of the matrix.

Decide which of these matrices are nilpotent.

• (a)  $\begin{pmatrix}-2&4\\ -1&2\end{pmatrix}$     (b)  $\begin{pmatrix}3&1\\ 1&3\end{pmatrix}$     (c)  $\begin{pmatrix}-3&2&1\\ -3&2&1\\ -3&2&1\end{pmatrix}$     (d)  $\begin{pmatrix}1&1&4\\ 3&0&-1\\ 5&2&7\end{pmatrix}$     (e)  $\begin{pmatrix}45&-22&-19\\ 33&-16&-14\\ 69&-34&-29\end{pmatrix}$

Consider the matrix from Example 2.19.

• Use the action of the map on the string basis to give the canonical form.

• Find the change of basis matrices that bring the matrix to canonical form.

• Use the answer in the prior item to check the answer in the first item.

Describe the effect of left or right multiplication by a matrix that is in the canonical form for nilpotent matrices.

Is nilpotence invariant under similarity? That is, must a matrix similar to a nilpotent matrix also be nilpotent? If so, with the same index?

Show that the only eigenvalue of a nilpotent matrix is zero.

Is there a nilpotent transformation of index three on a two-dimensional space?

In the proof of Theorem 2.16, why isn’t the proof’s base case that the index of nilpotency is zero?

Let $t\colon V\to V$ be a linear transformation and suppose $\vec{v}\in V$ is such that $t^{k}(\vec{v})=\vec{0}$ but $t^{k-1}(\vec{v})\neq\vec{0}$. Consider the $t$-string $\langle\vec{v},t(\vec{v}),\dots,t^{k-1}(\vec{v})\rangle$.

• 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.

• Prove that the restriction is nilpotent.

• Prove that the $t$-string is linearly indepe…

Finish the proof of Theorem 2.16.

Show that the terms ‘nilpotent transformation’ and ‘nilpotent matrix’, as given in Definition 2.7, 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 matri…

Recall that similar matrices have the same eigenvalues. Show that the converse does not hold.

Lemma 2.1 shows that any for any linear transformation $t\colon V\to V$ the restriction $t\colon\mathscr{R}_{\infty}(t)\to\mathscr{R}_{\infty}(t)$ is one-to-one. Show that it is also onto, so it is an automorphism. Must it be the identity map?

Prove that a nilpotent matrix is similar to one that is all zeros except for blocks of super-diagonal ones.

Prove that if a transformation has the same range space as null space. then the dimension of its domain is even.

Prove that if two nilpotent matrices commute then their product and sum are also nilpotent.