Products of real matrices with prescribed characteristic polynomials (Q2784373)

Let \(A\) be an \(n\times n\) matrix with entries in a field \(F.\) The authors propose to find necessary and sufficient conditions under which \(A\) is a product of two matrices \(B\) and \(C\) with prescribed characteristic polynomials \(f\) and \(g,\) respectively. They obtain a number of partial results for arbitrary fields \(F.\) If \(F\) is the field of real numbers and if \(n\geq 3,\) they show that \(A=BC\) if and only if \((fg)(0)=\det A\) and \(x^{n-\text{ rank}A}\) divides \(fg\).