I am familiar with Rayleigh Ritz Ratio for hermitian matrices. Let $A\_1$ be a given $N \times N$ hermitian matrix. Then the smallest eigenvalue of $A\_1$ is given by
\begin{align}
\lambda\_{min}(A\_1)=\min\_{x^Hx=1}x^HA\_1x
\end{align}
This led me to the following question. Let me define a new quantity 
\begin{align}
\lambda\_{min}(A\_1,A\_2)=\min\_{x^Hx=1}\max(x^HA\_1x,x^HA\_2x)
\end{align}
where $A\_2$ is another given hermitian matrix. Is this quantity known before?. Is any there intuitive way of looking at it? 


I was able to figure out that I could solve it using semi-definite optimization. Any comments on what happens when I try it for $k$ matrices $A\_1,A\_2,\ldots,A\_k$ 



Rayleight Ritz Ratio and smallest eigenvalue for a set of given matrices
Here is a partial answer. But let me first modify the question by replacing the reals by the complex. Hence, $x$ runs over ${\mathbb C}^n$ and $A\_1,A\_2$ are Hermitian. Then $A\_1,A\_2$ can be viewed as the "real" and "imaginary" parts of a single matrix $M=A\_1+iA\_2$, that is
$$A\_1=\frac12(M+M^\*),\qquad A\_2=\frac1{2i}(M-M^\*).$$
The advantage of working within the complex numbers is that we know (Toeplitz-Hausdorff Theorem) that the image ${\cal H}(M)$ of the unit sphere of ${\mathbb C}^n$ under the numerical map $x\mapsto x^\*Mx=x^\*A\_1x+ix^\*A\_2x$ is a convex compact subset of the complex plane.


Now, $\lambda\_\min(A\_1,A\_2)$ (under the complex definition) is the minimum of the convex function $f(z)=\max(\Re z,\Im z)$ over ${\cal H}(M)$. We easily see that it is achieved on the boundary of ${\cal H}(M)$, at a point which is either the downmost point, or the leftmost point, or the most SW point along the diagonal $\Delta$ defined by $\Re z=\Im z$.


Specifically, we have the following alternative, where $x$ (resp. $y$) denotes some unit eigenvector of $A\_1$ (resp. $A\_2$) associated with $\lambda\_\min(A\_1)$ (resp. $\lambda\_\min(A\_2)$) :


* either $x^\*A\_1x>x^\*A\_2x$, and then $\lambda\_\min(A\_1,A\_2)= \lambda\_\min(A\_1)$,
* or $y^\*A\_2y>y^\*A\_1y$, and then $\lambda\_\min(A\_1,A\_2)= \lambda\_\min(A\_2)$,
* or $x^\*A\_1x\le x^\*A\_2x$ **and** $y^\*A\_2y\le y^\*A\_1y$, and then $t:=\lambda\_\min(A\_1,A\_2)$ is the number such that $(t,t)$ is the down-left point of the interval ${\cal H}(M)\cap\Delta$.


