According to Dai(1989, p. 234), the following system:
$Ex(k+1)=Ax(k)+Bu(k)$
$y(k)=Cx(k), $ $k=0, 1, ..., L$
where $ x(k) \in \mathbb{R}^n$, $ u(k) \in \mathbb{R}^m$, $y(k) \in \mathbb{R}^r$ and $E, A \in \mathbb{R}^{n\times n}$, $B \in \mathbb{R}^{n\times m}$, $C \in \mathbb{R}^{r\times n}$ with $Rank(E)<n$
is causal if and only if $deg(det(zE-A))=Rank(E)$
Proof. (Necessity) From definiton 8-1.1 and equation 8-1.6 from Dai(1989), it can be proved that if the above system is causal then $deg(det(zE-A))=Rank(E)$; but I have troubles trying to prove the sufficiency. Any suggestion?