Factoring scalar damping from a matrix evolution

Statement

Let $U(t,s)$ be the propagator of $z'=A(t)z$ , and let $d(t)$ be a continuous scalar. The propagator for $z'=(A(t)-d(t)I)z$ is

V(t,s)=\exp\left(-\int_s^t d(r)\,dr\right)U(t,s).

Differentiating this product proves the identity and the initial condition $V(s,s)=I$ . No commutation between $A(t)$ at different times is needed, because the damping factor is scalar.

Harmonics

If $V_m$ corresponds to $A-m^2dI$ , then

V_m(t,s)=\exp\left(-(m^2-1)\int_s^t d(r)\,dr\right)V_1(t,s).

For $d\ge0$ , $t\ge s$ , and $|m|\ge1$ , higher harmonics have no larger propagator norm than the first harmonic under this comparison.