$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\si}{\sigma} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\theta} \newcommand{\om}{\omega} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathcal{F}} \newcommand{\E}{\operatorname{\mathsf E}} \newcommand{\Var}{\operatorname{\mathsf Var}} \renewcommand{\P}{\operatorname{\mathsf P}} \newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}} \newcommand{\eD}{\overset{\text{D}}\to} \newcommand{\D}{\overset{\text{D}}=} \newcommand{\tsi}{\tilde\si}$

This question is a follow-up to the previous question on symmetric matrices. Thanks to the responses by Christian Remling and Geoff Robinson to that question, the problem now becomes much more specific, as follows.

Suppose that a symmetric matrix $M\in\R^{n\times n}$ is orthogonal or, equivalently, satisfies the condition $M^2=I_n$, where $I_n$ is the $n\times n$ identity matrix. Suppose also that all the diagonal entries of $M$ are equal to one another. Is it then true that $M$ is of the form $aI_n+b\,\de\de^T$, where $\de$ belongs to the set (say $\De_n$) of all $n\times 1$ column matrices with all entries from the set $\{-1,1\}$; $a\in\{-1,1\}$; and $b\in\{0,-2a/n\}$?

This is true for $n\in\{2,3\}$.

Comment 1. Let $\mathcal M_n$ denote the set of all matrices $M$ satisfying the stated conditions, that is, the set of all symmetric orthogonal matrices $M\in\R^{n\times n}$ with constant diagonal entries.
The actual problem here is to show that for each $M\in\mathcal M_n$ all the off-diagonal entries $M_{ij}$ of $M$ with $i\ne j$ are of the form $c\de_i\de_j$ for some real $c$ and some $\de\in\De_n$; it is then easy to specify the appropriate $a$ and $b$, given in the above question. So, it is enough to show that \begin{equation} \prod_{\de\in\De_n}\Big(\sum_{1\le i<j\le n}\sum_{1\le k<\ell\le n}(M_{ij}\de_i\de_j-M_{k\ell}\de_k\de_\ell)^2\Big)=0, \end{equation} which is how the cases the cases of $n\in\{2,3\}$ were verified.

Comment 2. More generally, even without the condition on the diagonal entries of $M$, it is enough to show that $M-aI_n$ is of rank $1$ for some real $a$.

  • $\begingroup$ what is an $n\times n$ column matrix? $\endgroup$ – Abdelmalek Abdesselam Jul 20 '18 at 15:35
  • 2
    $\begingroup$ @AbdelmalekAbdesselam : Thank you for spotting the typo. It is now corrected. $\endgroup$ – Iosif Pinelis Jul 20 '18 at 15:36

No. A symmetric $M$ will satisfy $M^2=1$ if and only if the spectrum is contained in $\pm 1$, which is equivalent to $M=P-(1-P)=2P-1$ for some orthogonal projection $P$. Now you're asking if the extra condition that the diagonal is constant will give $P$ rank $1$ or $n-1$.

It's clear that this won't follow because we have such examples with $\textrm{rank }P=1$ for $n=2$ and can just take orthogonal sums of those for larger (even) $n$'s.

| cite | improve this answer | |
  • $\begingroup$ So, the cases of $n\in\{2,3\}$ are the only ones when the answer is yes. Turning to orthoprojectors is a nice idea. $\endgroup$ – Iosif Pinelis Jul 20 '18 at 17:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.