Positive Semidefinite Matrices

Question

Let $x=\left[ \begin{array}{cccc} x_{1} & x_{2} & \cdots & x_{n}% \end{array}% \right] $ be a vector with $\sum x_{i}=1$ and $x_{i}>0$. Is there an easy way to prove that \begin{equation} \mathrm{diag}\left\{ x_{1},x_{2},\ldots ,x_{n}\right\} -xx^{T} \end{equation} is positive semi-definite and has one zero eigenvalue with algebraic dimension one.

Answer 1

Try using the definition of semi-positive ($v^*Av\geq0$), the fact that all $x_i\in[0,1]$ (you need both bounds!) and Young's inequality ($2ab<a^2+b^2$) for $n$ terms. A 0-eigenvector (which is a slick way to say solution to the homogeneous system) is perhaps written in front of you?