A note on matrices with prescribed off-diagonal submatrix and characteristic polynomial

Let A∈Fq×p be a nonzero matrix over a field F, and let f be any monic polynomial of degree n=p+q with coefficients in F. A completion problem proposed and solved by de Oliveira asks for the existence of a matrix of order n with characteristic polynomial f and such that its q×p submatrix in the left down corner is equal to A. Let C(f)∈Fn×n be the companion matrix of f. We will construct a matrix XA∈Fn×n (which depends exclusively on A) such that XA−1C(f)XA is a closed form solution of this completion problem.