I have attached a small snippet from my lecture notes where we are dealing with constrained optimization. $E,C$ are matrices.
I am confused with Eq. (4.12) as to how they have differentiated with respect to $E$ when $E^T$ is present. Then, I do not know how they get from Eq. (4.12) to Eq. (4.13).
I am assuming that $C$ is symmetric.
Let $\phi(E) = E^T CE -\lambda E^T E$, then $\phi(E+H) = \phi(E)+H^TCE+ E^TCH -\lambda H^T E-\lambda E^T H +r(H)$, where $\|r(H)\| \le K \|H\|^2$.
Then $D \phi(E)H = H^T(CE - \lambda E) + ( CE - \lambda E)^T H$.
Suppose $D \phi(E)(H) = 0$ for all $H$, then choose $H=CE - \lambda E$ to get $2 (CE - \lambda E)^T (CE - \lambda E) = 0$ from which we get $CE - \lambda E = 0$ (since $A=0$ iff $A^TA = 0$).
I believe the derivation given in the question is incorrect, or at least misleading.