Does subtracting a rank-1 matrix from a positive definite matrix result in positive semi-definite matrix?

Question

Let $A$ be a symmetric positive definite matrix. We define the matrix $B$ as follows:

\begin{equation} B = A - \frac{1}{e^\top A e} Aee^\top A \end{equation}

where $e=(1,1,...,1)$.

I believe that $B$ is positive semi-definite, but I am unable to prove it.

Answer 1

By the Cholesky decomposition there exists a matrix $L$ such that $A=LL^\top$. Then, $$v^\top B v = v^\top A v - \frac{(v^\top A e)^2}{e^\top A e} = \|L^\top v\|^2 - \frac{((L^\top v)^\top (L^\top e))^2}{\|L^\top e\|^2}.$$ Now use Cauchy-Schwarz to show that this quantity is nonnegative.