The matrices I discuss are all $N\times N$ hermitian matrices. Consider two (hermitian) matrices $A_1$ and $A_2$. For a real scalar $t$, define the following function for the matrix $A_1+t*A_2$ \begin{align} \lambda(t)=\min_{u^Hu=1}~u^H(A_1+t*A_2)u \end{align} For a given $t$, this is essentially the smallest eigenvalue of $A_1+t*A_2$. This should be a concave function. Now define the function \begin{align} f(t)=||A_1+t*A_2||_F^2 \end{align} I randomly generated two hermitian matrices in my simulation software (matlab), and plotted this for a range of $t$ . You can see something happening here. $f(t)$ (blue curve) appears to be a convex function. The main thing being the value of $t$ at which $f(t)$ attains its minimum and $\lambda(t)$ (red curve) attains its maximum seems to be same. Is there any explanation for this. I tried it several times (more than 15 times at least), the nature of the resulting graph is always same. Can anyone give a possible explanation for this.
UPDATE--- I checked the exact values. They are close. But not the same. But still any possible explanation for this behaviour.