Convexity of the Joint Numerical Range
Chi-Kwong Li, Yiu‐Tung Poon · SIAM Journal on Matrix Analysis and Applications · 2000
Let A=(A 1 , . . ., A m ) be an m-tuple of n × n Hermitian matrices. For $1 \le k \le n$, the k{\rm th} joint numerical range of A is defined by $$W_k(A) = \{ ({\rm \tr}(X^*A_1X), \dots, {\rm \tr}(X^*A_mX) ): X \in {\bf C}^{n\times k}, X^*X = I_k \}.$$ We consider linearly independent families of Hermitian matrices {A 1 , . . . , A m } so that W k (A) is convex. It is shown that m can reach the upper bound 2k(n-k)+1. A key idea in our study is relating the convexity of W k (A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized numerical ranges and the corresponding matrix construction problems.