A constant polynomial matrix is a polynomial matrix whose elements are constant polynomials, i.e. polynomials with no indeterminates. Constant polynomials are obtained by "lifting" a "scalar" (i.e. an element of the underlying ring) into the polynomial ring/algebra by a "scalar injection", i.e. applying the "algebra scalar injection function" (see df-ascl) to a scalar : . In an analogous way, constant polynomial matrices (over the ring ) are obtained by "lifting" matrices over the ring by the function (see df-mat2pmat), called "matrix transformation" in the following.
In this section it is shown that the set of constant polynomial x matrices over the ring is a subring of the ring of polynomial x matrices over the ring (cpmatsrgpmat) and that is a ring isomorphism between the ring of matrices over a ring and the ring of constant polynomial matrices over the ring (see m2cpmrngiso). By this, it is shown that the ring of matrices over a commutative ring is isomorphic to the ring of scalar matrices over the same ring, see matcpmric. Finally , the transformation of a constant polynomial matrix into a matrix, is the inverse function of the matrix transformation , see m2cpminv.