Proof that $x x^T$ is symmetric and positive semidefinite

Question

For a generic vector $x\in\mathbb{R}^n$ it is clear the outer product $xx^T$ is a symmetric matrix.

But how can we prove that this is a positive semidefinite as well?

The matrix $X:=xx^T$ is given by the elements $x_{ij} = x_ix_j$.

A reference or answer would be great.

Answer 1

Let $Y$ be a column vector $\in\Bbb R^n$ so, since $Y\cdot x$ is a scalar, $$Y^TXY = Y^Txx^TY = (x^TY)^T(x^TY)\implies Y\cdot XY = (x^TY)^2 \geq 0.$$