A remark on the definition of the characteristic polynomial for a coupled system of ordinary differential equations (Q2703973)

A coupled system of differential equations is considered: NEWLINE\[NEWLINE\dot x(t)= Ax(t)+ Bu(t),\quad y(t)= Cx(t),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\dot z(t)= Rz(t)+ Qg(t)+ Ly(t),\quad u(t)= Kz(t)+ Q'g(t)+ L'y(t).NEWLINE\]NEWLINE The coefficients are assumed to be constant matrices of appropriate sizes, the unknown functions are vector valued. Using the Binet-Cauchy formula, an explicit representation is derived for the characteristic polynomial of the coupled system (after Laplace transform). The representation involves polynomial terms only, and is constructed using the characteristic polynomials of \(A\) and of \(R\) and subdeterminants of the polynomial matrices \(C(\text{adj}(sI- A))B\) and \(K(\text{adj}(sI- R))L+ L'\text{det}(sI- R)\).