Convex hull is a subset of the numerical range

Question

Question Show that if A is a normal matrix, then W(A) (the numerical range of A) is the convex hull of the eigenvalues of A. Progress I have proved that the W(A) ⊆ conv A similar to Numerical range of a normal matrix is the convex hull of its eigenvalues. But I am unsure of how to prove that the conv A ⊆ W(A). I have started with a convex combination and sort of am trying to work backwards from what I have and letting v=(Sqrt(t1), ..., Sqrt(tn)), but I do not know how to finish this.

Answer 1

Let $l = \sum_k t_k \lambda_k $ where $t_k \ge0 $ and $\sum_k t_k = 1$. Since $A$ is normal, we can write $A=U \Lambda U^*$ where $U$ is unitary and $\Lambda$ is diagonal. Let $x=U(\sqrt{t_1},...,\sqrt{t_n})$, then $x^* A x = l$.