A short proof of the elliptical range theorem

Abstract

A short proof of the elliptical range theorem concerning the numerical range of $2\times 2$ complex matrices is given.

The basic properties of the numerical range were uncovered by Toeplitz [4] and Hausdorff [2]. A key feature is the elliptical range theorem of $2\times 2$ complex matrices due to Toeplitz. We present a proof in the spirit of Toeplitz and Davis [1], but more explicitly, including the lengths of the axes (cf. Uhlig [5]), in presentation more like as of Li [3].

FormalPara Theorem 1

If A is a $2\times 2$ complex matrix,  then its numerical range

$$\begin{aligned} {\textrm{W}}(A)=\{\langle A{\textbf{x}},{\textbf{x}}\rangle \,:\, {\textbf{x}}\in {\mathbb {C}}^2,\, |{\textbf{x}}|=1 \} \end{aligned}$$

is a possibly degenerate elliptical disk on the complex plane, with the eigenvalues of A as the foci,  and with

$$\begin{aligned} s^{\pm }(A)=\frac{1}{2}\sqrt{ {{\,\textrm{tr}\,}}\,\left( A-\frac{{{\,\textrm{tr}\,}}A}{2}{{\,\textrm{Id}\,}}\right) ^*\left( A-\frac{{{\,\textrm{tr}\,}}A}{2}{{\,\textrm{Id}\,}}\right) \pm \left| {{\,\textrm{tr}\,}}\,\left( A-\frac{{{\,\textrm{tr}\,}}A}{2}{{\,\textrm{Id}\,}}\right) ^2\right| } \end{aligned}$$

as the major and minor semi-axes.

FormalPara Proof

Applying the transform $A\mapsto {\textrm{e}}^{i\theta }A+v{{\,\textrm{Id}\,}},$ $\theta \in {\mathbb {R}},$ $v\in {\mathbb {C}},$ it transforms the range and the eigenvalues accordingly, while $s^{\pm }(A)$ are left invariant. Also, conjugation by a unitary matrix leaves all these data invariant. By this, we can assume that $A=\begin{bmatrix} c&{}2b\\ 0&{}-c\end{bmatrix}$ such that $b,c\in [0,+\infty )$. Then, for ${\textbf{x}}=\begin{bmatrix} z_1\\ z_2\end{bmatrix}$, $|z_1|^2+|z_2|^2=1$, it yields

$$\begin{aligned} \begin{bmatrix} {{\,\textrm{Re}\,}}\langle A{\textbf{x}},{\textbf{x}}\rangle \\ {{\,\textrm{Im}\,}}\langle A{\textbf{x}},{\textbf{x}}\rangle \\ 0 \end{bmatrix}= & {} \begin{bmatrix} b&{}&{}-c\\ {} &{}b&{}\\ {} &{}&{}0 \end{bmatrix} \begin{bmatrix}2{{\,\textrm{Re}\,}}(\bar{z}_1z_2) \\ 2{{\,\textrm{Im}\,}}(\bar{z}_1 z_2)\\ |z_2|^2-|z_1|^2\end{bmatrix} \nonumber \\= & {} \underbrace{\begin{bmatrix} \sqrt{b^2+c^2}&{}&{}\\ {} &{}b&{}\\ {} &{}&{}0\end{bmatrix}}_F \underbrace{\begin{bmatrix} \frac{b}{\sqrt{b^2+c^2}}&{}&{}\frac{-c}{\sqrt{b^2+c^2}} \\ {} &{}1&{} \\ \frac{c}{\sqrt{b^2+c^2}}&{}&{}\frac{b}{\sqrt{b^2+c^2}}\end{bmatrix}}_R \underbrace{\begin{bmatrix}2{{\,\textrm{Re}\,}}(\bar{z}_1z_2) \\ 2{{\,\textrm{Im}\,}}(\bar{z}_1z_2)\\ |z_2|^2-|z_1|^2\end{bmatrix}}_S \end{aligned}$$

(1)

(in the case of $b=c=0$, any orthogonal matrix can be chosen for R).

Taking all unit vectors ${\textbf{x}}$, S ranges over the unit sphere (the base of the Hopf fibration). Applying R leaves it invariant. Applying F independently dilates in the first and second coordinates, and totally contracts in the third one. Thus (1) (third coordinate omitted) ranges over a possibly degenerate elliptical disk of canonical position with major semi-axis $\sqrt{b^2+c^2}$ and minor semi-axis b. Its foci are then $(\pm c,0)$. Now, these data are according to the statement of the theorem. $\square$