Rank-Based Norms, Capra-Conjugacies and the Rank Function

Abstract: 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.