Consider two $N \times N$ hermitian indefinite matrices $A\_1$ and $A\_2$. Consider their affine combination 
\begin{align}
M(t)=(1-t)A\_1+tA\_2
\end{align}
I am interested in the minimum eigenvalue of $M(t)$. I can write this as 
\begin{align}
\lambda(t)=\min\_{x ~\in~\mathbb{C}^{N\times 1}}~&x^HM(t)x \\\
&x^{H}x = 1
\end{align}


When I simulated this, I made the following observations


* It is a concave function of $t$
* It is a piece-wise linear function of $t$ (at least, that is what the plot looks like).


**My question, 1) how do I explain this behavior 2) Any comments on $t$ where the maxima occurs?**


Some Thoughts:


I have a feeling that this is connected to the Toeplitz-Hausdorff theorem. From it, it follows that the subset of 2-D plane 
\begin{align}
\mathbb{S}=\{[x\_1,x\_2]\in \mathbb{R}^2\mid ~x\_1=x^HA\_1x,~x\_2=x^HA\_2x,~x^{H}x=1\}
\end{align} 
is a closed compact convex subset of $\mathbb{R}^2$. Thus, it follows that minimum eigenvalue of $M(t)$ is 
\begin{align}
\lambda(t)=(1-t)x\_1+tx\_2, ~~[x\_1,x\_2]\in \mathbb{S}
\end{align}
Now from here (if this is correct?), How do I conclude the observations, I made earlier? 



Minimum eigenvalue of a Affine Combination of two Hermitian matrices
This is related to so-called *hyperbolic polynomials*, studied by L. Gaarding in the fifties. More generally, let $\lambda(\xi)$ be the least eigenvalue of $A(\xi)=\sum\_\alpha\xi\_\alpha A^\alpha$, where $A^\alpha$ are Hermitian matrices and $\xi$ is a real vector. Then $\lambda$ is a concave function. It is generically strictly concave, except in the radial directions of course because of homogeneity $\lambda(s\xi)=s\lambda(\xi)$ for $s>0$. The strict concavity is related to the lack of commutativity of pairs $(A^\alpha,A^\beta)$. For instance, the Pauli matrices yield $\lambda(\xi)=-|\xi|$, which is clearly strictly concave away from rays ${\mathbb R}^+\xi$. On the opposite, if $[A^\alpha,A^\beta]=0$ for every pair, then $\lambda$ is piecewise linear.


Strict concavity occurs for instance when the least eigenvalue is simple for every $\xi\ne0$, or if it has a constant multiplicity. Then $\lambda$ is analytic away of the origin, with a Hessian matrix of rank $n-1$. It turns out that this property implies a so-called *Strichartz inequality* for the solutions of the system of Partial Differential Equations
$$\frac{\partial u}{\partial t}+\sum\_\alpha A^\alpha\frac{\partial u}{\partial x\_\alpha}=0.$$ 
Obviously, the symbol of the system is $\det(\tau I\_n+A(\xi))$.Thus the characteristic manifold is related to the eigenvalues of $A(\xi)$, in particular to $\lambda(\xi)$.


The other eigenvalues satisfy more involved inequalities such as Weyl, Lidskii, Ky Fan -type inequalities. For instance, the sum of the $k$ least eigenvalues is concave too. This is a part of Alfred Horn's conjecture, now a theorem thanks to the work of many people, including Knutson & Tao.


The last part of the question, that about Toeplitz-Hausdorff theorem, is unclear. ${\mathbb S}$ is not a singleton, so what means 
\begin{align}
\lambda(t)=(1-t)x\_1+tx\_2, ~~[x\_1,x\_2]\in \mathbb{S} \qquad ?
\end{align}
The equality certainly holds true for teh particular point $(x\_1,x\_2)$ obtained by taking $x$ a unit eigenvector associated with $\lambda(t)$, but what else ? To see some deep relations between Toeplitz-Hausdorff and hyperbolic polynomials, have a look to our paper with Th. Gallay, *Numerical measure of a complex matrix*, in
Comm. Pure Appl. Math. **65** (2012), no. 3, 287–336. 


