Given an invertible matrix $A$ and column vectors $b$ and $c$.

For which $A$,$b$ and $c$ are all corresponding principal minors of $B = A-bc^T$ and $A^{-1}$ equal?

According to a result by Loewy, this is true if $B$ and $A^{-1}$ are diagonally similar with transpose (plus some extra conditions). This work by Engel and Schneider seems more promising, but still I'm stuck. We can assume that both $A$ and $A^{-1}$ are adjacency matrices of fully connected graphs, i.e., all entries are non-zero.

My main interests are:

1. For which matrices $A$ is the problem solvable?
2. Given a matrix $A$, how attain $b$ and $c$ numerically (best possible if no exact solution exist)?

Motivation: The problem arises in control theory, where transfer function from a state-space formulation is:

$$ H(z) = \frac{\det(A) \det(D(z) - (A - bc^T))}{\det(D(z) - A)}, $$ where $D(z) = diag([z^{m_1},\dots,z^{m_n}])$ for integer $m_i$. The goal is now to choose $A$, $b$ and $c$ such that $|H(z)|=1$ for all $z$. This is true if the numerator and denominator of $H(z)$ are "flipped", i.e.,

$$ flip(\det(D(z) - A)) = \det(A) \det(D(z) - A^{-1}). $$

Thus, for any $m_i$, we need: $$ det(D(z) - A^{-1}) = det(D(z) - (A - b*c^T)), $$ which is true if all principal minors are equivalent.