The Implementation part of an Axiom capsule for a domain constructor uses the special variable to Rep @ Rep identify the lower level data type used to represent the objects representation:of a domain of the domain. domain:representation The for quadratic forms is SquareMatrix(n, K). This means that all objects of the domain are required to be by matrices with elements from K.
The code for quadraticForm in Figure figquadform checks that the matrix is symmetric and then converts it to $, which means, as usual, ``this domain.'' Such explicit conversions conversion are generally required by the compiler. Aside from checking that the matrix is symmetric, the code for this function essentially does nothing. The m :: $ on line 28 coerces to a quadratic form. In fact, the quadratic form you created in step (3) of ugDomainsDemo is just the matrix you passed it in disguise! Without seeing this definition, you would not know that. Nor can you take advantage of this fact now that you do know! When we try in the next step of ugDomainsDemo to regard as a matrix by asking for nrows, the number of its rows, Axiom gives you an error message saying, in effect, ``Good try, but this won't work!''
The definition for the matrixmatrixQuadraticForm function could hardly be simpler: it just returns its argument after explicitly coercing its argument to a matrix. Since the argument is already a matrix, this coercion does no computation.
Within the context of a capsule, an object of $ is regarded both as a quadratic form and as a matrix.In case each of $ and have the same named operation available, the one from $ takes precedence. Thus, if you want the one from Rep, you must package call it using a $Rep suffix. This makes the definition of easy---it just calls the dotdotDirectProduct product from DirectProduct to perform the indicated operation.