Determining asymptotic stability using transfer function?
Poles in the right half-plane only cause instability if the system is a continuos-time system, because in that case, a pole $s_{\infty}$ contributes a signal

$$k\cdot e^{s_{\infty}t}u(t)\tag{1}$$

to the output signal, with some constant $k$, and with $u(t)$ denoting the unit step function. For $(1)$ to decay we require $\text{Re}\{s_{\infty}\}&lt;0$, i.e., for a (causal) continuous-time system to be stable, all its poles must lie in the left half-plane.

In discrete time, however, a pole $z_{\infty}$ contributes a sequence

$$k\cdot z_{\infty}^nu[n]\tag{2}$$

to the output signal. Consequently, in order for $(2)$ to decay, we require $|z_{\infty}|&lt;1$, i.e., for a (causal and) stable discrete-time system, all poles must be inside the unit circle.

Concerning the equivalence of poles and eigenvalues of the system matrix have a look at this answer.