In this paper we introduce
a sharpening of the Parikh mapping and investigate
its basic properties.
The new mapping is based on square
matrices of a certain form. The classical Parikh vector
appears in such a matrix as the second diagonal.
However, the matrix product gives more information about
a word than the Parikh vector. We characterize the matrix
products and establish also an interesting interconnection
between mirror images of words and inverses of .