The characteristic polynomial of a matrix with prescribed off-diagonal blocks (Q5961574)

Let \(F\) be a field, \(A_{1,2}\in F^{p\times q}\), \(A_{2,1}\in F^{q\times p}\) and \(f(x)\) be a monic polynomial over \(F\) of degree \(p+q\). The following problem has not yet been solved, although some partial answers are known: NEWLINENEWLINENEWLINEProblem. Under what conditions do there exist matrices such that (1) \(A=\left[\begin{smallmatrix} A_{1,1} &A_{1,2}\\ A_{2,1} &A_{2,2} \end{smallmatrix} \right]\) has characteristic polynomial \(f(x)\)? NEWLINENEWLINENEWLINEThe following theorems give new sufficient conditions to answer the problem. In Theorem 2, the authors generalize a theorem of \textit{F. C. Silva} [Port. Math. 44, 261-264 (1987; Zbl 0656.15002)], with an almost direct proof. NEWLINENEWLINENEWLINETheorem 1. If \(q=1\), then there exist matrices \(A_{1,1}\in F^{p\times p}\) and \(A_{2,2}= [a]\), \(a\in f\), such that the matrix (1) has characteristic polynomial \(f(x)\). NEWLINENEWLINENEWLINETheorem 2. Suppose that \(f(x)= f_1(x)f_2(x)\), where \(f\) has degree \(p\). If one of the following conditions is satisfied, then there exist \(A_{1,2}\in F^{p\times p}\), \(A_{2,2}\in F^{q\times q}\) such that the matrix (1) has characteristic polynomial \(f(x)\): NEWLINENEWLINENEWLINE(i) \(t>1\), (ii) \(t=1\) and \(p\neq q\), (iii) \(t=1\), and \(p\neq q\), (iii) \(t=1\), and one of the polynomials \(f_1(x)\) or \(f_2(x)\) is reducible.