Chapter 7.1, Problem 23E

Prove property (b) of Theorem $2$.

THEOREM $2$ Let $q\left(x\right)$ be a quadratic form with representation $q\left(x\right)=x^{T}Ax$, where $A$ is a symmetric $\left(n\times n\right)$ matrix. Let the eigenvalues of $A$ be ${\lambda }_1$, ${\lambda }_2$, …, ${\lambda }_n$. The quadratic form is:

(b) Positive semidefinite if and only if ${\lambda }_i\ge 0$, for $1\le i\le n$.