If you look at a z-transform table it might express z-transforms in terms of $z^{-1}$. Some also in terms of $z$. They are always expressed as negative. i.e.) $-z^{-1}$
Here is my question, how does the signal change if it is $z^{-1}$, i.e.) positive? Is it correct to assume that the sign of the corresponding signal would change?
For example, the signal: $a^{n}u(n)$ has the z-transform, $$\frac{1}{1-az^{-1}}$$
Would, $-a^{n}u(n)$ have the z-transform, $\frac{1}{1+az^{-1}}$???
Thanks for insight
Assuming $z \neq 0 $ (necessary anyway since the function has a pole at zero), the transform is equivalent to $\frac{z}{z-a}$ which is a standard case with inverse $au(n)$. $z$ is just a complex algebraic variable and can be manipulated as such.