Some Results on Fields of Values of a Matrix
Daniel Gries, Josef Stoer · SIAM Journal on Numerical Analysis · 1967
1. For a square matrix A the so-called field of values G[A] is defined as the following set of complex numbers:' (1.1) G[A] := {xHAx|xHx = 1}. It is well known that this set contains the spectrum A(A) = {Xj(A) } of A. Moreover, Toeplitz [11] proved in 1918 that G[A] also contains the convex hull 5C( A(A)) of A(A). Hausdorff [5] generalized this result by proving that G[A] is convex. The concept of a field of values was generalized by Bauer [1] in 1962. Starting with an arbitrary norm || v 1l in Cn (or R') and its dual norm (defined on the dual space of Cn (or Rn) of row vectors yH) IIyH D =max Re y X