  We consider the space of matrices, with given number of rows and of columns,
equipped with the classic trace scalar product. With any matrix (source) norm,
we associate a coupling, called Capra, between the space of matrices and
itself. Then, we compute the Capra conjugate and biconjugate of the rank
function. They are expressed in function of a sequence of rank-based norms,
more precisely generalized r-rank and dual r-rank matrix norms associated with
the matrix source norm. We deduce a lower bound of the rank function given by a
variational formula which involves the generalized r-rank norms. In the case of
the Frobenius norm, we show that the rank function is equal to the variational
formula.
