Suppose I know all eigenvalues of a symmetric matrix $A$, and also of all its principal submatrices.
Given symmetric matrix $B$ that is similar to $A$ -- under which circumstances will also all principal submatrices of $B$ be similar to the corresponding ones of $A$?
An obvious condition seems that their eigenvectors coincide, which implies $B = A$. Can one get away with a weaker condition?
I assume that a principal submatrix of $A$ is obtained removing some rows and columns of same indices.
The required result for $n=3$: let $A=\begin{pmatrix}a&d&e\\d&b&f\\e&f&c\end{pmatrix}$; then the convenient matrices $B$ are
$B=\begin{pmatrix}a&\epsilon_1d&\epsilon_2e\\\epsilon_1d&b&\epsilon_3f\\\epsilon_2e&\epsilon_3f&c\end{pmatrix}$, where $\epsilon_i=\pm 1$ and $(\epsilon_1\epsilon_2\epsilon_3-1)dfe=0$.