  Single-input eigenvalue assignment problem (SEVAS) for dense (non-sparse)
weakly controllable pairs A,B using Ackermann's formula is revisited.
Factorizations are presented that are interpreted using quotient (factor)
vector spaces. Depending on the representations chosen for the equivalence
class (given as specific projections by real orthogonal matrices), the
numerical behavior of pole placement scheme can be enhanced. One version
operates by placing one pole at a time (with complex conjugate poles grouped
together to avoid complex arithmetic). Another version operates with the
coefficients of the characteristic polynomial directly. The latter version uses
orthogoanl real matrices. In both cases, there are no constraint on the number
of identical poles. A version of the algorithm uses only ring arithmetic. The
algorithms are compared with numerically stable algorithms that appeared in the
literature, such as the Miminis-Paige algorithm or the Varga pole shifting
method. In case of real distinct eigenvalues to be placed, a geometrical
interpretation as the interesection of affine hyperplanes provides the value of
the gain vector, which seems a novel interpretation.
