Real Positive Semidefinite Matrices
Definition
Let $A \in \mathbfsf{S} ^n$ (i.e., $A \in
\R ^{n \times n}$ and symmetric).
$A$ is positive definite
if
\[
x^\top A x > 0 \quad \text{for all nonzero } x \in \R ^d
\] \[
x^\top A x \geq 0 \quad \text{for all } x \in \R ^d
\]
Notation
We denote the set of positive definite $d$ by $d$ matrices by $\mathbfsf{S} _{++}^d$. We denote the set of positive semidefinite $d$ by $d$ matrices by $\mathbfsf{S} _{+}^d$.