Symmetric, Positive Semidefinite, and Positive Definite Real Matrices

Let $k\ge1$ be a natural number and let $A$ be a real $k\times k$ matrix.

$A$ is symmetric if $A=A^{\top}$, with the transpose.

A symmetric $A$ is positive semidefinite if, with the dot product on the Euclidean space $\mathbb{R}^{k}$ and the matrix-vector product,

and positive definite if $x\cdot(Ax)>0$ for every nonzero $x\in\mathbb{R}^{k}$.