The following sequence space is presented. Let $[t]$ be a positive double sequence and define the sequence space $\Omega''(t)=\{\text{complex sequences }x:x_{k,l}=O(t_{k,l})\}$. The set of geometrically dominated double sequences is defined as $G''=\bigcup_{r,s\in(0,1)}G(r;s)$, where $G(r,s)=\{\text{complex sequences }x:x_{k,l}=O(r^k,s^l)\}$ for each $r,s$ in the interval $(0,1)$. Using this definition, four dimensional matrix characterizations of $l^{\infty,\infty}$, $c''$ and $c''_0$ into $G''$ and into $\Omega''(t)$ are presented. In addition to these definitions and characterizations it should be noted that this ensure a rate of converges of at least as fast as $[t]$. Other natural implications are also presented.