Characteristic polynomials and controlability of partially prescribed matrices

A note on the controllability of pairs of matrices

“…(See [1].) Let F be an arbitrary field and let n, p 1 , p 2 , p 3 be positive integers such that n = p and C 1,2 has full rank, there is no guarantee of the existence of a completely controllable pair of the form (2) with the given blocks.…”

“…There are many results available in the literature that describe the possibility that a system of the form (1) be completely controllable, when some entries of [A|B] are prescribed. In particular, in a previous paper [2] we described conditions under which there exists a completely controllable pair of the form…”

Controllability of partially prescribed matrices

“…In [2] we established conditions under which there exists a completely controllable pair of the form (5), when k − 1 blocks of the same size are prescribed and the others are unknown.…”

“…Later, in [2] we described the possible characteristic polynomials of a matrix of the form (3) partitioned into k × k blocks of the same size p × p, with entries in an arbitrary field F. Our answer shows that it is always possible to prescribe k − 1 blocks of the matrix and the characteristic polynomial, except if all the nonprincipal blocks of a row or column are prescribed equal to 0 and the characteristic polynomial has not any divisor of degree p.…”

Matrices with prescribed eigenvalues and prescribed submatrices

“…Still considering all the blocks C i,j of size p  p, in [4] we showed that it is possible to prescribe k À 1 blocks of (2), simultaneously with the characteristic polynomial. Notice that the result established by Dias da Silva in [7] can be obtained as a particular case of this, with p ¼ 1.…”