Page 29 - Textos de Matemática Vol. 44

NATA´LIA BEBIANO’S LEGACY 19
Theorem. Write the matrix A = H + iG with hermitian matrices H and G. If ux + vy + w = 0 is the equation of a support line of VS(A) then
det(uH + vG + wS) = 0.
Proof. We sketch the interesting proof. Write Re(.) for referring to the real part of a square matrix, possibly a complex number. As is the case for the classical numerical range, for every real (angle) φ, one has that
VS(Re(e−iφA)) = Re(VS(e−iφA)) ⊆ R.
By p-convexity this set is either R itself or a segment or its complement in R. We collect in a set Ω the angles φ for which we do not get R. For φ ∈ Ω let zj = zj(e−iφA), j = 1,2, be the endpoints of the set VS(Re(e−iφA)); then the lines x = zj are support lines of VS(Re(e−iφA)). Hence the by φ rotated lines, which are cos φ x + sin φ y = zj (e−iφA), are the supportlines of the rotated figure; i.e. of eiφVS(e−iφA) = VS(A). By ranging with φ over Ω we get all support lines. Now by a theorem of Li and Rodman, since zj(e−iφA) is a corner of Re(VS(e−iφA)), it is an eigenvalue of
S−1 Re(e−iφA) = cos φS−1H + sin φS−1G.
The equality here follows by a little calculation. Now if μ is an eigenvalue of
such a matrix, it satisfies
det(cos φ S−1H + sin φ S−1G − μIn) = 0.
From this one gets the claim.
To get such explicit descriptions of the boundary of VS,C(A) is certainly difficult, but the authors manage to obtain via a theorem of Tarski from real algebraic geometry the following theorem.
Theorem. Let c = (γ1,...,γn) ∈ Cn and A = diag(α1,...,αn). Then the boundary of VS,c(A) is a finite union of algebraic arcs and hence a curve of class C∞ except for a number of points.
As is well known, if A is a 2 × 2 matrix with eigenvalues λ1 and λ2 then the classical numerical range W(A) is an ellipse with foci λ1,λ2 and minor axis
tr(A∗A) − |λ1|2 − |λ2|2.
The authors establish on pages 211-219 a strikingly similar theorem for the set VJ (A).