I am trying to learn the fundamentals of numerical range (or field of values). My apologies in advance for asking such questions that might be trivial for you. But I don't know yet how to solve them.
If $A = \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ and $B = \begin{bmatrix}0 & 2 \\ 0 & 0\end{bmatrix}$, then
how to determine $F(A),\ F(B)$ and $F(A + B)$?
How to show that $F(A+B)$ is a proper subset of $F(A) + F(B)$.
Thank you so much in advance for your kind help.
p.s.: Definition of $F(A) ≡ \{x^∗Ax : x \in \mathbb{C}^n, x^∗x = 1\}$ (cf. [1] C. R. Johnson and R. Horn, "Topics in Matrix Analysis".)
$F(A)$ is clearly the line segment $[1,2]$. $F(B)$ can be easily shown to be the closed unit disc.
$F(A+B)$ is more tricky. Fortunately, it is known that the numerical range of every $2\times2$ matrix is an ellipse (including the degenerate case where the ellipse collapses to a line segment) and there are explicit formulae for the two axes. See Charles R. Johnson, Computation of the Field of Values of a $2\times2$ Matrix, Journal of Research of the National Bureau of Standards - B. Mathematical Sciences, vol. 78B, no. 3, July-September, 1974.
In your case, $F(A+B)$ is an ellipse whose major axis is $\left[\frac{3-\sqrt{5}}2,\,\frac{3+\sqrt{5}}2\right]$ and whose minor axis is $[-i,i]$.