A Note on a Theorem of Forsythe and Golub

Source abstract

Forsythe and Golub (1965) proved a theorem about all stationary points of the function $\Phi ( x ) = ( {x - b} )^H A( {x - b} )$ for x on the sphere $x^H x = 1$. The same theorem is proved using much simpler methods. It is also shown how the result can be used to find the minimum and maximum of $\Phi ( x )$ for $x^H x\leqq 1$. It turns out that, with a few exceptions, the minimum and maximum occur on the boundary $x^H x = 1$.