  In this paper, we describe the Grothendieck groups $K_1(V)$ and $K(V)$ of an
absolute matrix order unit space $V$ for unitary and partial unitary elements
respectively. For this purpose, we study some basic properties of unitary and
partial unitary elements, and define their path homotopy equivalence. The
construction of $K(V)$ follows in a almost similar manner as that of $K_1(V).$
We prove that $K_1(V)$ and $K(V)$ are ordered abelian groups. We also prove
that $K_1(V)$ and $K(V)$ are functors from the category of absolute matrix
order unit spaces with morphisms as unital completely $\vert \cdot
\vert$-preserving maps to the category of ordered abelian groups. Later, we
show that under certain conditions, quotient of $K(V)$ is isomorphic to the
direct sum of $K_0(V)$ and $K_1(V),$ where $K_0(V)$ is the Grothendieck group
for order projections.
