Let $\mathcal{S}_n$ denote the vector space of all real symmetric $n \times n$ matrices.
Is there a characterization of the subspaces $V$ of $\mathcal{S}_n$ that are invariant under orthogonal similarity transformations i.e. $QVQ^T = V$ for all orthogonal $n \times n$ matrices $Q \in O(n)$?
The only example I can think of is $V = \{ X \in \mathcal{S}_n : \text{tr}(X) = 0 \}$ since conjugation does not change the set of eigenvalues of a matrix.
Are there other examples of such subspaces?
A partial answer: every subspace with codimension $1$ (i.e. of dimension $\dim(S_n) - 1$) can be described as $$ V_A = \{X \in S_n : \operatorname{tr}(AX) = 0\} $$ If $A$ is such that $V_A$ is invariant under orthogonal conjugation, then for every orthogonal $U$, we have $$ \operatorname{tr}(AX) = 0 \implies \operatorname{tr}(AUXU^T) = \operatorname{tr}(U^TAUX) = 0 $$ To put it another way: for all $X$: either $\operatorname{tr}(AX) = 0$, or $\operatorname{tr}([U^TAU]X) = 0$ for all $U$. I have a hunch that this will only hold if $A$ is a multiple of an identity, which is to say that the subspace you found is the only such subspace of codimension $1$.