Symmetric and Orthogonal Matrices – Properties of a Matrix That is Both Symmetric and Orthogonal

Question

I tried to find matrices A, which are both orthogonal and symmetric, this means
$A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix
$$\begin{pmatrix}
0 &0& -1\\
0& -1& 0\\
-1& 0& 0
\end{pmatrix} $$
Can a matrix with the desired properties only contain the values -1,0 and 1 ?
Which matrices of a given size have the desired property ?

Answer 1

For your first question, the answer is no. Every real Householder reflection matrix is a symmetric orthogonal matrix, but its entries can be quite arbitrary.

In general, if $A$ is symmetric, it is orthogonally diagonalisable and all its eigenvalues are real. If it is also orthogonal, its eigenvalues must be 1 or -1. It follows that every symmetric orthogonal matrix is of the form $QDQ^\top$, where $Q$ is a real orthogonal matrix and $D$ is a diagonal matrix whose diagonal entries are 1 or -1.