Is matrix with entries $a_{ij} = x_ix_j$ positive semidefinite?

Question

I wonder if I can proof that the square matrix with entries $a_{ij} = x_ix_j$ for a given vector $x$ is positive semidefinite.

I hope it is because this matrix is somehow related to this question where I asked if mean square error is convex function in linear regression. I actually calculated the Hessian matrix and obtained the one I give you (multiplied by 2). The problem is that I don't know how to proof that it is positive semidefinite so that I can show that mean square error is convex. Please give a proof or a counterexample.

Answer 1

You can easily check that $A=xx^T$. Then, for any $y$, $$ y^TAy=y^Txx^Ty=(x^Ty)^Tx^Ty\geq0. $$

Answer 2

If $x=(x_1,x_2,\dots,x_n)$ is the row vector, then $A=x^Tx$. So give a column vector $y$, you have that $y^TAy=(y^Tx^T)(xy)=(xy)^T(xy)$. What is that never negative?