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?
Best Answer
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.