Entry-wise square of orthogonal matrix

Question

Let $O(n)$ be the space of all $n\times n$ orthogonal matrix. Consider the following set: $$S_n=\{(U_{ij}^2)_{1\leq i,j\leq n}: U\in O(n)\}$$ Then it is obvious that $S_n$ is a subset of the space of all doubly stochastic matrices (every row and column sums to 1). My question is that does every doubly stochastic matrix have a representation in terms of the entry-wise square of an orthogonal matrix? In other words, does $S_n$ equal the space of all doubly stochastic matrices?

Answer 1

Only if $n=2$. For $n\geq3$, there are counterexamples. The most famous one is $$\tfrac12\begin{bmatrix}1&1&0\\0&1&1\\1&0&1\end{bmatrix}.$$

Answer 2

A dimensionality argument suffices to disprove this. The group of orthogonal real matrices $O(n,\mathbb R)$ has dimension $n(n-1)/2$, while the set of doubly-stochastic matrices has dimension $(n-1)^2$. The freedom to add a finite number of $\pm$ signs does not change the dimension, so it is impossible for orthogonal matrices to cover all doubly-stochastic matrices once $n/2 > 1$.