Functional Equations in the Theory of Dynamic Programming. Ix: Variational Analysis, Analytic Continuation, and Imbedding of Operators.

only if a = ZFi xi -xi*. We shall derive a somewhat stronger form of this result. We first prove THEOREM 3. Let a be positive. Then the nonzero characteristic roots of Ra are all positive. For we have already shown that we may restrict our attention to the case where = [a]. By hypothesis, 6(xax) > 0 for every x of I, and so we see that 6(xax) = xaSx' = XRaSX' > 0 and the symmetric matrix RaS of equation (10) is positive semidefinite. If x is a characteristic vector of Ra and a is a characteristic root such that xRa = ax, we have S(xax) axSx'. But xSx' > 0, xRaSX' > 0 for all real nonzero vectors xi, and so axSx' > 0 a > 0 as desired. We now derive the final result. THEOREM 4. Let 2I be an involutorial algebra over the field 9? of all rational numbers, and let 6(x) be a functional satisfying relation (4). Then an element a = a* of 21 is positive if and only if a = x12 + x22 + x32 + x42for xi = xi* in 9R[a]. For J? [a I is semisimple and so may be written as the direct sum 9? [a] = T [a,ID . [at] for algebraic fields 9?[ai] over R such that a = a, + ... + at. If 4 4 t ai= xij2 then a = xj2, where xj = xij. Also, 6(xiaxi) = 6(xiaixi) > 0 j=1 j=1 i=1 for every xj of 9?[aJ], 6(xi2) > 0 for every xi # 0 of 9?[ai]. Thus we have reduced the proof of our theorem to the case where 2f = 9? [a] is a totally real algebraic number field. Since a is totally positive, the quadratic form f(y) = y12 + y22 + y12 + y42 ay62 has the property that all algebraic number conjugates are indefinite. By a theorem of H. Hasse,2 the form f(y) is a null form, that is, y12 + y22 + y32 + y42 = ay52 for yi not all zero and in 9?[a]. But yI 5 0, since otherwise yi2 + ... + y42 = 0 for real yi not all zero. Hence a = X12 + X22 + X32 + x42, with xi = Yj(y5)1.