Fields of values of a matrix

for some vector x, where (x, x)1z=1x1+ * ** x,x. It is known that F,(A) is a closed bounded convex region containing the characteristic values X1, * * *, Xn of A and with these points as foci of the algebraic curve bounding F,(A). A summary of results and a bibliography has recently been given by Parker [1]1 who observes that several authors have given bounds for the roots which are actually bounds for F,(A). Since F,(A) is invariant when A is replaced by B = UA U-1 for U unitary but may be altered if B = TA T-1 for arbitrary nonsingular T, information about F,(A) cannot be expected to give precise information on the Xi. Evidently the difficulty is that F,(A) depends not only on the linear mapping described by A but also involves a choice of metric which is unrelated to the linear mapping. If, however, we define for each positive definite hermitian matrix H, a field of values FH(A) consisting of all complex numbers