Suppose $A$ is a $k\times k$ symmetric matrix with $A_{ij}>0$ for all $i,j$ and $\sum_{i,j} A_{ij} = 1.$ Let $A_i$ be the sum of elements in row $i$ (or column $i$) of $A.$
Let $B$ be a matrix with entries $$B_{ij} = \frac{A_{ij}^2}{A_i A_j}.$$ Let $B^\prime$ be identical to $B$ except that its row $k$ and column $k$ are zeroed out.
Let $$\|C\|:=\sup_{u:\|u\|=1}\|Cu\|$$
be the operator norm induced by the Euclidean distance.
I want to show a quantitative version of the following statement:
If $A_i$ is large $(\sim \Theta\left(\frac 1k\right))$ for all $i$ except $i=k,$ then $\frac{\|B^\prime\|}{\|B\|}$ is very close to 1, and approaches $1$ as $A_k$ approaches zero.
Note that the entry $B_{kk}$ need not be small, although yes, $B_{ik}, B_{ki}$ for $i\neq k$ will be small.
Note also that I am interested in the ratio of the two norms being close to 1, not their difference being close to 0.
