It is shown that for $n \le 3$ the joint numerical range of a family commuting $n\times n$ complex matrices always convex; \ge 4$ there are two whose not convex.

Let A be an n× n normal matrix, whose numerical range NR[A] is a k -polygon. If a unit vector v ∈ W ⊆ Cn, with dimW = k and the point v∗Av ∈ IntNR[A], then NR[A] is circumscribed to NR[P∗AP], where P is an n× (k− 1) isometry of {span{v}}⊥ W → Cn, [1]. In this paper, we investigate an internal approximation of NR[A] by an increasing sequence of NR[Cs] of compressed matrices Cs = R∗s ARs, with R∗...