Constructing matrices with prescribed main-diagonal submatrix and characteristic polynomial

Let F be any field. Let A11 be a matrix of Fp×p and let f be a monic polynomial of F[x] of degree p + q. A result of de Oliveira characterizes the conditions for the existence of matrices A12∈Fp×q, A21∈Fq×p and A22∈Fq×q such that f is the characteristic polynomial of the 2 × 2 block matrix . Ikramov and Chugunov obtained a finite step algorithm to construct A12, A21 and A22 when they exist. We present an algorithm that clearly simplifies the previous one.