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Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Suppose that $A_1,\dots,A_r\in M_n(K)$ are all Hermitian. Define a function $f_{A_1,\dots,A_r}:\mathbb{RP}^{n-1}\rightarrow[0,\infty)$ by setting $$f_{A_1,\dots,A_r}([x_1,\dots,x_r])=\frac{\|x_1A_1+\dots+x_rA_r\|_\infty}{\|(x_1,\dots,x_r)\|_2}.$$

Here, $\|A\|_\infty=\sup\{\|Ax\|_2:\|x\|_2=1\}$ is just the usual spectral norm. For the quaternionic case, one can just associate each Hermitian matrix in $M_n(\mathbb{H})$ with a Hermitian matrix in $M_{2n}(\mathbb{C})$.

Does $f_{A_1,\dots,A_r}$ have at most $n$ local maximum values $f_{A_1,\dots,A_r}([x_1,\dots,x_r])$? If not, then how many local maximum values can $f_{A_1,\dots,A_r}$ have?

I have been searching for counterexamples, and while I have been able to get $n$ local maxima, I have not been able to get $n+1$ local maxima. For example, let $e_1,\dots,e_n$ be an orthonormal basis for $K^n$, and set $A_j=e_je_j^\ast$ for $1\leq j\leq n$. Let $\mathbf{x}_j=(\delta_{1,j},\dots,\delta_{n,j})$. Then $[\mathbf{x}_1],\dots,[\mathbf{x}_n]$ are $n$ distinct local maxima for $f_{A_1,\dots,A_n}$.

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  • $\begingroup$ I originally had a more general conjecture, but I was able to use equal norm tight frames to refute this more general conjecture. Maybe equal norm tight frames or something like that can be used here too. $\endgroup$
    – Joseph Van Name
    Commented Jun 1, 2023 at 12:21
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    $\begingroup$ Are the $A_i$ suppose to be Hermitian/symmetric? In the question, you write $A_j \in M_n(K)$, so no symmetry hypothesis, but then you talk about Hermitian matrices later. Interesting question! $\endgroup$
    – David E Speyer
    Commented Jun 1, 2023 at 13:19

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