Suppose $A$ is a positive definite matrix such that$$ I \preceq A \preceq 1.01I.$$ Is it possible that $\sum\limits\_{i=1}^n A\_{1i}$ can be arbitrarily large?



Possible pathological properties of positive definite matrix
Outline of one possible answer (but am doing this in a rush so have not written anything down to check).


Consider the matrix $\Gamma$ which has 0 in top-left corner, 1 in rest of top row and rest of 1st column, and 0 in all other entries. (Note to hovering LaTeX hawks: please don't feel the need to enclose all those numbers in dollars.)


Clearly $\Gamma$ is hermitian and a hasty calculation seems to show its eigenvalues are $\pm\sqrt{n-1}$ and 0.


Take $A= I + 0.005 (I+ (n-1)^{-1/2}\Gamma )$. This ought to satisfy your sandwich condition but it is clear that the sum of entries in 1st column will be ${\rm O}(\sqrt{n}$).


