Positive semidefinite

A square, symmetric matrix $H ∈ \mathbb{R}^{d×d}$ is positive semidefinite (PSD) for any vector $y ∈ \mathbb{R}^d$, $\mathbf{y}^\intercal \mathbf{Hy} ≥ 0$.

Notation

Use “Loewner order” \succeq to denote $\mathbf{H}$ is PSD, i.e. $$\mathbf{H} \succeq 0$$ Write $\mathbf{B} \succeq \mathbf{A}$ or equivalently $\mathbf{A} \preceq \mathbf{B}$ to denote that $(\mathbf{B}-\mathbf{A})$ is PSD; this gives a partial ordering on matrices.

Convexity and PSD

If $f$ is twice differentiable, then it is convex if and only if the matrix $∇^2 f(\mathbf{x})$ (Hessian matrix) is positive semidefinite for all $\mathbf{x}$.