We introduce a new P basis for the Hopf algebra of quasisymmetric functions that refine the symmetric powersum basis. Its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. The product and coproduct are then extended to matrix fillings thereby defining a Hopf algebra of matrix fillings. We lift our quasisymmetric powersum P basis to the Hopf algebra of quasisymmetric functions in non-commuting variables by introducing fillings with disjoint sets. Finally, we look at this P basis under Hivert's local action.