Eigenvalues and Definiteness
Why
Can we characterize positive (semi-)definite matrices in terms of their eigenvalues?
Main Result
Using eigenvalue decompositions, we can answer in the affirmative.
Suppose $A \in \mathbfsf{S} ^d$ has smallest
eigenvalue $\lambda _{\min}(A)$.
Then
\[ \begin{aligned} A \in \mathbfsf{S} _+^d \quad &\iff \quad \lambda _{\min}(A) \geq 0 \\ \quad &\iff \quad \tr AB \geq 0 \text{ for all } B \in \mathbfsf{S} _+^d. \end{aligned} \]
and\[ \begin{aligned} A \in \mathbfsf{S} _{++}^d \quad &\iff \quad \lambda _{\min}(A) > 0 \\ \quad &\iff \quad \tr AB > 0 \text{ for all nonzero } B \in \mathbfsf{S} _{++}^d. \end{aligned} \]