  The purpose of this paper is to introduce and study quadratic $n$-ary
Hom-Nambu algebras, which are $n$-ary Hom-Nambu algebras with an invariant,
nondegenerate and symmetric bilinear forms that are also $\alpha$-symmetric and
$\beta$-invariant where $\alpha$ and $\beta$ are twisting maps. We provide
constructions of these $n$-ary algebras by using twisting principles, tensor
product and T*-extension. Also is discussed their connections with
representation theory and centroids. Moreover we show that one may derive from
quadratic $n$-ary Hom-Nambu algebra ones of increasingly higher arities and
that under suitable assumptions it reduces to a quadratic $(n-1)$-ary Hom-Nambu
algebra.
