On a special form of the Cauchy-Schwarz inequality (Q1073135)

In 1955 R. Bellman published the following result: if T is a complex nonsingular matrix, then \((g,(TT^ h)g)\cdot (f,(TT^ h)^{-1}f)\geq (f,g)(g,f)\) for all complex vectors f and g (here \((x,y)=x^ hy\) is the inner product of x and y, \(T^ h=(T^*)^ t)\). In the present paper the author refines this result. Namely, he finds all complex nonsingular matrices T for which the function \((T^{-1}f,T^{-1}f)(T^ hg,T^ hg)\) reaches its minimal value \(| f^ hg|^ 2\). A similar problem is solved for real matrices T (f and g are still complex) where the minimum is max \(\{| f^ hg|^ 2,| f^ tg|^ 2\}\). As an application the author considers the problem of minimization of parameter sensitivity in state-space systems.