  We propose a specific class of matrices which participate in factorization
problems that turn to be equivalent to constant and entwining (non-constant)
pentagon, reverse-pentagon or Yang-Baxter maps, expressed in non-commutative
variables. In detail, we show that factorizations of order $N=2$ matrices of
this specific class are equivalent to the homogeneous normalization map. From
order $N=3$ matrices, we obtain an extension of the homogeneous normalization
map, as well as novel entwining pentagon, reverse-pentagon and Yang-Baxter
maps.
