Positive Semidefinite and Positive Definite Matrices by Their Eigenvalues

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $n\in\mathbb{N}$ and $A\in\mathcal{S}(n)$, the set of symmetric matrices, with positive semidefinite and positive definite matrices, eigenvalues, invertible matrices, the matrix-vector product, the dot product and the Euclidean norm $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$.

¶ $A$ is positive semidefinite if and only if every eigenvalue of $A$ is $\ge0$.

¶ The following are equivalent: $A$ is positive definite; every eigenvalue of $A$ is $>0$; $A$ is positive semidefinite and invertible.

¶ If $A$ is positive definite, then there is a real number $c>0$ with $x\cdot(Ax)\ge c\lVert x\rVert^{2}$ for every $x\in\mathbb{R}^{n}$.