Impulse Matching: Finding the Unit Impulse Response

[Corneo]

Impulse Matching

In regards to finding the unit impulse response of a system. We assume that $x(t) = \delta (t)$ and that the intials conditions at $t=0^_$ are all zero. The impulse response $h(t)$ therefore must consists of the systems's modes for when $t \geq 0^+$. But why is it that $h(t) = A \delta(t) + \text{modes}$ for $t \geq 0$?

 

[phantom_photon]

I guess the simplest explanation would be that if the domain of the response is extended to include t = 0 (or 0-) it will have an additional component: the system's response to the impulse while the impulse is being applied, which is of course a scalar multiple of the Dirac delta function.
Since the delta function is non-zero for t = 0- and 0 but not for 0+, the response for t >= 0+ will not involve the delta function at all, just it's residual effect (the system's characteristic oscillations). You will generally only be concerned with the response for t > 0+.
That's what I think it is, but I might be wrong. I appologise for the lack of mathematical rigour in this post.