On prescribed characteristic polynomials (Q6618702)

Assume that \(\mathbb{F}\) is a field and \(\mathcal{M}_n(\mathbb{F})\) denote the matrix ring consisting of all square matrices of size \(n \times n\) over \(\mathbb{F}\). For a monic polynomial \N\[\Nq(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0\in\mathbb{F}[x] \N\]\Nthe coefficient \(-a_{n-1}\in\mathbb{F}\) is called trace of \(q(x)\). If such a polynomial is the characteristic polynomial of a certain matrix, the trace coincides with the trace of such a matrix.\N\NLet \(\mathbb{F}\) be a field, \(n, k\in\mathbb{N}\) with \(k<n-k\), and consider the block matrix \N\[\NA=\left( \begin{array}{cc} \mathbf{0}_{k, k} & \mathbf{0}_{k, n-k}\\\N\mathbf{0}_{n-k, k} & A_{22} \end{array} \right)\in \mathcal{M}_n(\mathbb{F}),\N\]\Nconsisting of \(k\) rows and columns of zeros and an invertible non-derogatory matrix \(A_{22}\). The authors prove that for any monic polynomial \(q(x)\) of degree \(n\) whose trace coincides with the trace of \(A\), there exists a square-zero matrix \(N\) such that the characteristic polynomial of \(A+N\) coincides with \(q(x)\).