Vertex points in the numerical range of a derivation
Marvin D. Marcus, Markus Sandy · Linear and Multilinear Algebra · 1987
Let A be an n-square normal matrix with eigenvalues λ1, …, λ n . For 1 ⩽ r ⩽ m < n let Qm,n denote the strictly increasing sequences of integers ω satisfying For arbitrary indeterminates t 1, …tm denotes the rth elementary symmetric function of t 1, …tm . The set Hr,m is the convex hull of the complex numbers If B is an m-square complex matrix, then define Er (B) by where is the r-square principal submatrix of B lying in rows and columns numbered α. Define Am by The main theorem is: Let 1 ⩽ r ⩽ m < n and let A be an n-square complex normal matrix with eigenvalues λ1, …, λ n . Then Er (Am )εHr,m . Assume that Er (λ1, …, λ n .) is a vertex of Hr,m , that Er (Am ) = Er ( λ1, …, λ m ), and that Er (λ1, …, λ m ) ≠ Er (λω(1), …, λω(m)), ω ≠ ϵ. Then A is a direct sum and the eigenvalues of Am are λ1, …, λ m .