How to show that this matrix is positive semi-definite?

Question

Let $x\in R^n$ and symmetric matrix $A\in R^{nxn}$ be defined as $$A_{ij} = x_ix_j$$ for i,j = 1,2,...,n

How can I show that this matrix is positive semi-definite? Any suggestions would be greatly appreciated.

Thank you!

Answer 1

$A = x x^T$ (where $x$ is considered as a column vector). Thus for any $y \in \mathbb R^n$, $y^T A y = y^T x x^T y = (x^T y)^2 \ge 0$.