There are a lot of well known theorems that relate the eigenvalues of a real symmetric matrix $A$ to the eigenvalues of its principal submatrices. I can't help but wonder, are there similar theorems which relate the eigenspaces in these cases? For instance, if $v$ is an eigenvector for the smallest eigenvalue of a principal submatrix, then can we say anything about the extension of $v$ in relation to the eigenspaces of $A$? I was hoping for something like:
If $v$ is an eigenvector for the smallest eigenvalue of a principal submatrix $B$ inside $A$, then $(v,0)$ is contained in the span of the two smallest eigenspaces of $A$. I.e. at worst, you need to use the smallest eigenvalue bigger than the eigenvalue associated to $v$ to describe $(v,0)$.
Sorry for being so imprecise, I am frankly unsure of what specific questions one can reasonably ask here.
