Numerical range of $2\times 2$ matrices Let $(A_1,\cdots, A_d)\in \mathcal{L}(\mathbb{C}^2)^d$. Consider
$$W(A_1,\cdots, A_d)=\{(\langle A_1 x,x\rangle,\cdots,\langle A_d x,x\rangle):x \in E,\;\;\|x\|=1\}.$$

If $A_k$ are commuting, why $W(A_1,\cdots, A_d)$ is convex?? 

 A: First off, if $A$ and $B$ are two commuting $2\times 2$ matrices which are no multiples of the identity matrix $I$, then there exist $\alpha,\beta \in \mathbb{C}$ such that $A=\alpha B+\beta I$. 
WLOG we can choose $k\in\lbrace 1,\ldots,d\rbrace$ such that $A_k$ is not a multiple of the identity (else the respective coordinate of $W$ is constant hence obviously convex). Then $A_j=\alpha_j A_k+\beta_j I$ for all $j\in\lbrace 1,\ldots,d\rbrace$ and thus
$$
W(A_1,\ldots,A_d)=(\alpha_1,\ldots,\alpha_d) W(A_k)+(\beta_1,\ldots,\beta_d)
$$
which yields the desired result as the numerical range of one operator is known to be convex.
A: For nxn matrices A_1, ..., A_d, one may assume that they are in diagonal form
with A_j = diag(a_j1, ..., a_jn). Then it is easy to verify that for any
unit vector x = (x_1, .., x_n)^t, 
(xA_1x, ..., xA_dx) is a convex combination of the points: 
(a_11, ..., a_d1), (a_12, ...., a_d2), ..., (a_1n, ..., a_dn).
One can then check that W(A_1, ..., A_d) is the convex hull of 
the above points and hence a convex polyhedron.
