The field of values bound on ideal GMRES

A widely known result of Howard Elman, and its improvements due to Gerhard Starke, Michael Eiermann and Oliver Ernst, gives a bound on the (worst-case) GMRES residual norm using quantities related to the field of values of the given matrix and of its inverse. In this note we give a simple and direct proof that these bounds also hold for the ideal GMRES approximation. Our work was motivated by a question of Otto Strnad, a student at the Charles University in Prague.