Is the spectral norm of $I_n-1v^T$ bounded by $\sqrt{n}$? Let $I_n$ be $n$-dimensional identity matrix and $v$ be a stochastic vector, i.e., $v$ is non-negative and $v^T{\bf 1}=1$, where ${\bf 1}=[1,1,...,1]^T$. I wonder if the spectral norm of the matrix $I_n-{\bf 1}v^T$ is bounded by $\sqrt{n}$.
It is easily seen that the spectral norm is bounded by $\sqrt{n}+1$ using the norm triangle inequality. However, I test several special cases such as $v_1=[1,0,..,0]^T$ and $v_2=[1,1,...,1]^T/n$ and the largest norm is $\sqrt{n}$ (attained at $v_1$). So I wonder if this bound can be made tighter.

You can check that $v \mapsto \lVert I-1v^T\rVert$ is convex using the definition of convexity and the properties of norms. The maximum of a convex function on the simplex is attained on one of its extremal points of the form $(0,\ldots,0,1,0,\ldots,0)$. This proves that your lower bound of $\sqrt{n}$ is actually tight.
