Best constants for the lp inequalities associated to some particular matrices

We compute the norm on lp of some special kind of matrices. The sharp computation (not just an estimate) of a matrix norm is a hard problem in general (see, e.g., [20]), but it turns out to be solvable for matrices having a suitable special form. This is analogous to what happens for some linear operators studied in analysis. Here we describe some families of matrices for which the solution can be found in a reasonably simple way. In particular, we focus on circulant matrices, on matrices having constant line sums, and on special kinds of block matrices. As a related result we can prove that any doubly stochastic matrix has p-norm equal to 1, which can be interpreted as an additional characterization of the Birkhoff polytope. This also leads to the sharp computation of the p-norm of special non-symmetric matrices where positive, zero or even negative entries appear.