In this evolving note, we record some algebra-flavored exercises relevant for the Oberwolfach seminar.
Let \(F\) be a field, let \(V\) be a finite-dimensional vector space over \(F\), and let \(M := \mathop{\mathrm{End}}(V)\) denote the space of linear maps \(V \to V\).
Definition 1. Let \(\tau \in M\) and \(v \in V\). We denote by \(F[\tau] v\) the set of elements of \(V\) that may be written as a polynomial in \(\tau\) applied to \(v\), or equivalently, the span of the elements \[v, \quad \tau v, \quad \tau^2 v, \quad (\dotsc).\] We say that a vector \(v \in V\) is \(\tau\)-cyclic, or that \(v\) is a cyclic vector for \(\tau\), if \[F[\tau] v = V.\] We say that \(\tau\) is cyclic if it admits a cyclic vector.
Exercise 1. Show that \(\tau\) is cyclic if and only if there is a basis with respect to which it is of the form, e.g., for \(\dim(V) = 4\), \[\begin{pmatrix} 0 & 0 & 0 & \ast \\ 1 & 0 & 0 & \ast \\ 0 & 1 & 0 & \ast \\ 0 & 0 & 1 & \ast \\ \end{pmatrix}.\]
Exercise 2. Show that the set of conjugacy classes consisting of cyclic elements is in bijection with the set of characteristic polynomials, that is to say:
For each monic polynomial of degree \(\dim(V)\), there exists a cyclic element \(\tau \in M\) whose characteristic polynomial is that polynomial.
Two cyclic elements with the same characteristic polynomial are conjugate.
Exercise 3. Show that a matrix given in Jordan form is cyclic precisely when the eigenvalues pertaining to different Jordan blocks are distinct. In particular, a diagonal matrix is cyclic precisely when its diagonal entries are distinct.
Definition 2. We say that \(\tau \in M\) is regular if \(\dim M_\tau = \dim V\), where \[M_\tau := \left\{ x \in M : x \tau = \tau x \right\}\] denotes the centralizer of \(\tau\) in \(M\).
Exercise 4. Show that for \(\tau \in M\), the following are equivalent.
\(\tau\) is cyclic.
\(M_\tau = F[\tau]\).
\(\tau\) is regular.
Definition 3. We recall that \(\tau \in M\) is nilpotent if some power of \(\tau\) vanishes.
Exercise 5. Suppose that \(F = \mathbb{R}\). Fix a norm \(\lVert . \rVert\) on \(M\). Let \[\mathcal{O} \subseteq M\] be a conjugacy class consisting of regular (equivalently, cyclic) elements. Let \(x_j \in \mathcal{O}\) be a sequence whose matrix norms \(\lVert x_j \rVert\) tend to infinity.
Show that, after passing to a subsequence if necessary, the normalized limit \[x := \lim_{j \rightarrow \infty } \frac{x_j}{ \lVert x_j \rVert}\] exists, and is nilpotent.
Show that for each \(\mathcal{O}\) as above, there exists a sequence \(x_j\) as above for which the normalized limit \(x\) is regular nilpotent.
Exercise 6. Suppose that a matrix of the form \[\tau = \begin{pmatrix} 0 & \ast & \ast & \ast \\ 0 & \ast & \ast & \ast \\ 0 & \ast & \ast & \ast \\ 1 & \ast & \ast & \ast \\ \end{pmatrix}\] has the property that the standard basis vector \(e_4\) is cyclic. Show that \(\tau\) is invertible.
Exercise 7. Show that if a matrix is cyclic, then so is its transpose.
Exercise 8. Let \(\mathcal{O}\) be a conjugacy class of \(n \times n\) matrices. Show that the intersection \[\mathcal{O}_{\psi} := \mathcal{O} \cap \begin{pmatrix} \ast & \ast & \ast & \ast \\ 1 & \ast & \ast & \ast \\ 0 & 1 & \ast & \ast \\ 0 & 0 & 1 & \ast \\ \end{pmatrix}\] is nonempty if and only if \(\mathcal{O}\) consists of regular (equivalently, cyclic) elements, in which case the group \[N := \begin{pmatrix} 1 & \ast & \ast & \ast \\ 0 & 1 & \ast & \ast \\ 0 & 0 & 1 & \ast \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}\] acts simply-transitively on \(\mathcal{O}_{\psi}\) via conjugation.