  In this paper, we will introduce a notion of basis related Morita equivalence
in the Cuntz--Krieger algebras $({{\mathcal{O}}_A}, \{S_a\}_{a \in E_A})$ with
the canonical right finite basis $\{S_a\}_{a \in E_A}$ as Hilbert
$C^*$-bimodule, and prove that two nonnegative irreducible matrices $A$ and $B$
are elementary equivalent, that is, $A = CD, B = DC$ for some nonnegative
rectangular matrices $C, D$, if and only if the Cuntz--Krieger algebras
$({{\mathcal{O}}_A}, \{S_a\}_{a \in E_A})$ and $({{\mathcal{O}}_B}, \{
S_b\}_{b\in E_B})$ with the canonical right finite bases are basis relatedly
elementary Morita equivalent.
