Definite triples of Hermitian matrices and matrix polynomials (Q1861314)

Given nonzero \(n\times n\) Hermitian matrices \(A\), \(B\), and \(C\), the joint numerical range of the triple \((A,B,C)\), that is, the set \(F(A,B,C)=\{(x^*Ax,x^*Bx,x^*Cx)\in \mathbb{R}^n: x\in \mathbb{C}^n,\;x^*x=1\}\) is convex for \(n\geq 3\) and, it is either convex or the surface of an ellipsoid for \(n=2\). The main result of this paper is a formula for the joint inner numerical radius of a general triple \((A,B,C)\), that is, the distance between \((0,0,0)\) and the boundary of \(F(A,B,C)\). An algorithm for testing definiteness and a study of the numerical ranges for special matrix polynomials are also included.