TITLE: The minimal norm of a shifted stochastic matrix
QUESTION [5 upvotes]: Given a row-stochastic matrix $M$ with singular values $\sigma_{1} \geq \cdots \geq \sigma_{n}$, I am looking for an upper bound on the expression
$$\min_{\alpha} \left\| M- \frac{\alpha}{n}J_{n} \right\|_{2}$$ 
where $J_{n}$ is the matrix with all ones.
It is not hard to see that if $M$ is doubly stochastic, the above expression is exactly $\sigma_{2}$ (as the singular vector of the largest singular values is the vector of all ones), for $\alpha =1$. Can you find a similar bound when $M$ is only row stochastic?
Thank you.
Edit: Suppose we take $a, b$ to be the left and right singular vectors corresponding to the largest singular value $\sigma_{1}$. Then, 
$$\| M- \frac{\alpha}{n}J_{n}-\sigma_{1}ab^T+\sigma_{1}ab^T \|_{2} < \sigma_{2}+\sigma_{1}\left(\sqrt{1-\frac{\left< a,e\right> ^{2}\left< b,e\right> ^{2}}{N^{2}}}\right)$$ 
for 
$$\alpha = \frac{\sigma_{1}\left< a,e\right> ^{2}\left< b,e\right>^{2}}{N^{2}}$$
For a doubly stochastic matrix, this bound is tight (as the first singular vectors are $\frac{1}{\sqrt{N}}e$). What can we say, for example, of $\left< a,e\right> \left< b,e\right>$, when $\sigma_{1}$ is not 1, but very close to it?

REPLY [3 votes]: Some experimentation led me to the following results:

$\alpha=1$ yields the minimum value even for the operator-2 norm (certainly provable directly too)
$\|M-ee^T/n\|_2$ can be larger than $\sigma_2 + \cdots + \sigma_n$
$\|M-2ee^T/n\|_2 = \sigma_1(M)$ (obviously, since $M$ is rs)

So the only reasonable bound is: $\|M-ee^T/n\|_2 \le c_n\sigma_1(M)$, where the constant $c_n$ depends on the dimensionality $n$. As of now, I am using $c_n=1$, but have not paid much thought to what the best constant would be (need to dig up some singular value inequalities for that).
Here is an example matrix for which $\|M-ee^T/n\|_2 \ge 0.9\sigma_1(M)$.
\begin{equation*}
 M = \frac{1}{1000}\begin{pmatrix}
   10  &  24  &  966\\\\
   410 &  576 &   14\\\\
   529 &  362 &  109
 \end{pmatrix}.
\end{equation*}