  We construct a new bigraded Hopf algebra whose bases are indexed by square
matrices with entries in the alphabet $\{0, 1, ..., k\}$, $k \geq 1$, without
null rows or columns. This Hopf algebra generalizes the one of permutations of
Malvenuto and Reutenauer, the one of $k$-colored permutations of Novelli and
Thibon, and the one of uniform block permutations of Aguiar and Orellana. We
study the algebraic structure of our Hopf algebra and show, by exhibiting
multiplicative bases, that it is free. We moreover show that it is self-dual
and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra,
we obtain a new one indexed by alternating sign matrices. We study some of its
properties and algebraic quotients defined through alternating sign matrices
statistics.
