Let $\lambda$ and $\Lambda$ be two fixed positive numbers and let $A$ be a symmetric real matrix with $\lambda |x|^2 \leq (Ax,x) \leq \Lambda |x|^2$.
Let $B$ be a matrix derived from $A$ as follows: $B$ retains the diagonal entries of $A$ and one off diagonal entry. An example would be $B$ having the diagonal and one above and below the diagonal entries from $A$ and the rest to be $0$.
Another example would be $B$ having diagonal and two above and below the diagonal entries from $A$. $$ \left(\begin{array}{ccccc}a_{11} & 0 & 0 & \ldots & 0\\ 0 & a_{22} & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & a_{nn} \end{array} \right), \left(\begin{array}{ccccc}a_{11} & a_{12} & 0 & \ldots & 0\\ a_{21} & a_{22} & a_{23} & \ldots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & a_{nn} \end{array} \right), \left(\begin{array}{ccccc}a_{11} & 0 & a_{13} & \ldots & 0\\ 0 & a_{22} & 0 & \ldots & 0 \\ a_{31} & 0 & a_{33} & \ldots & 0\\ \vdots & \ddots & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & a_{nn} \end{array} \right) $$
A bunch of such matrices can be constructed and are clearly symmetric. Is it also true that these matrices satisfy bounds of the form $\lambda |x|^2 \leq (Ax,x) \leq \Lambda |x|^2$ with the same $\lambda$ and $\Lambda$? Clearly the diagonal matrix does, but what about the others?
