Given $M\in M_n({\mathbb R})$ and $\ell\in{0,\ldots,n-1}$, we define $$d_\ell(M)=\sum_{j=1}^nm_{j,j+\ell},$$ where the indices are understood mod $n$. In particular, $d_0$ is the trace operator.

Let $A\in M_n({\mathbb R})$ be given. We define a map $\Delta: O_n({\mathbb R})\rightarrow{\mathbb R}^{n-1}$ by $$\Delta(R)=(d_1(R^TAR),\ldots,d_{n-1}(R^TAR)).$$ Mind that we omit $d_0(R^TAR)$, because we know in advance that it equals the trace of $A$.

**Question**. Does it exist an orthogonal matrix $R$ such that $\Delta(R)=(0,\ldots,0)$ ?

The requested property ressembles one for which the answer is known to be positive: find $R$ orthogonal such that the diagonal $R^TAR$ is constant (thus equal to $\frac{1}{n}{\rm Tr}A$). Both properties consist of $n-1$ linear constraints, and both are consistent with the fact that the mean value of $R^TAR$ over $SO_n$ is $(\frac{1}{n}{\rm Tr}A) I_n$. Thus the answer would certainly be positive if the stronger following statement is true.

**Statement**. The image of $SO_n$ under $\Delta$ is convex. True or False ?

This statement looks ambitious, since $\Delta$ is not linear, and $SO_n$ is not a convex set. But an optimistic mathematicien will say that it ressembles the Toeplitz-Hausdorff theorem about the convexity of the image of the complex unit sphere under the quadratic map $x\mapsto x^*Mx$. Note that the T-H thm is used to find an $R^TAR$ with constant diagonal.

the indices are understood mod n. Thus the functions $d_\ell$ are circular sums. There are $n$ terms in each sum. – Denis Serre Sep 29 '10 at 15:37your book;-) – Pietro Majer Sep 29 '10 at 16:20