Trace Minimization

In this case we consider a ``Rayleigh quotient'' style of iteration to find the eigenspaces of the smallest eigenvalues of a symmetric positive definite matrix $A$. That is, we can minimize $F(Y) = \frac{1}{2}\tr(Y^*AY)$ [14,183,392]. The minimizer $Y^*$ will be an $n\times p$ matrix whose columns span the eigenspaces of the lowest $p$ eigenvalues of $A$.

For this problem,

$$dF(Y) = AY,$$

it is easily seen that

$$\frac{d}{dt}dF(Y(t))\vert _{t=0} = AH,$$

where $\dot{Y}(0) = H.$