Let $B_{\bar{i}\bar{i}}$ denote the remnant of a square matrix $B$ after its $i^{th}$ row and $i^{th}$ column have been removed. Now given any vector $v$ is there some natural relation between the quantities $v'^T B_{\bar{i}\bar{i}} v'$ and $v^TBv$? (where $v'$ is such that it is $v$ with its component along the $i^{th}$ coordinate is removed)
Any inequality or an equation that can be written down amongst them?
particularly if $v$ has no component along the $i^{th}$ coordinate?
do kindly state if anything strikes your mind about these expressions! (even if it doesn't directly answer the question!)
It seems the following.
Let $b_i$ is the $i^{th}$ row of the matrix $B=\|b_{km}\|$ and $b^i$ is the $i^{th}$ column of the matrix $B$. Then $$v'^T B_{\bar{i}\bar{i}} v'=v^TBv-v_i(v,b_i^T)-v_i(v,b^{i}) +v_i^2b_{ii}.$$ In particular, if $v_i=0$ then $v'^T B_{\bar{i}\bar{i}} v'=v^TBv.$