Is there anything interesting about this matrix update? I have a symmetric, real matrix $M$ (not necessarily full rank) and I am performing the following rank one update $$M - \frac{M\textbf{1}\textbf{1}^TM}{\textbf{1}^TM\textbf{1}}$$ where $\textbf{1}$ is the vector of all ones. Is there anything interesting I can say about this matrix (properties, theorems that use a similar update, perturbation effects) or is it just a boring old rank one update?
 A: Among all symmetric matrices $A$ satisfying $A \mathbf{1}=0$, the matrix in your update is the "closest" to $M$ in some inverse sense. This is related to to the Hessian update step of the DFP numerical optimization algorithm*. 
Specifically, consider the following optimization problem**:
\begin{align}
\min_{\text{symetric matrices }A} &\quad ||A^{-1} - M^{-1}||^2_\text{Fro} \\
\text{such that} & \quad A x = b.
\end{align}
where $||\cdot||_\text{Fro}$ is the Frobenius matrix norm. The solution to this optimization problem is:
$$A = M - \frac{Mx x^T M}{x^T M x} + \frac{bb^T}{x^T b}.$$
Taking $x=\mathbf{1}$ and letting $b$ be some vector with very small but nonzero entries, $b=O(\epsilon)$, yields an optimal update of the form:
$$M - \frac{M\mathbf{1} \mathbf{1}^T M}{\mathbf{1}^T M \mathbf{1}} + O(\epsilon).$$
As $b \rightarrow 0$, this becomes the update formula in your post. 

*See section 6.1 of Numerical Optimization by Nocedal and Wright.
** If $M$ is not invertible, you can perturb it by some small amount to make it invertible, $M' = M + O(\epsilon)$ and all of the discussion in this post still goes through fine and you get the same results when $\epsilon \rightarrow 0$.
A: Clearly the matrix $$T = M - \frac{M\textbf{1}\textbf{1}^TM}{\textbf{1}^TM\textbf{1}}$$
has vector $\mathbf{1}$ as its eigenvector, corresponding to eigenvalue of zero:
$$
   T \cdot \mathbf{1} = M \mathbf{1} - \frac{\left(M \mathbf{1}\right) \left(\mathbf{1}^t M \mathbf{1}\right) }{\mathbf{1}^t M \mathbf{1}} = M \mathbf{1} - M \mathbf{1} = 0
$$
A: In statistics, if $M$ represents a contingency table of observed values, then $$E=\frac{M11^TM}{1^TM1}$$ represents the "expected values".
The row and column sums (or marginals) are the same for both $M$ and $E$. 
Also, $E$ would be used to calculate the Chi-Squared statistic for the observations.
Was that interesting?
