negative semidefinite matrix

Question

I got a positive definite matrix $B$, that is, $V(x)=x^TBx>0$ for any vector $x≠0$. I am clear with the statement that $λ_\min∥x∥_2^2≤V(x)≤λ_\max∥x∥_2^2$ for any $x≠0$, where $λ_\min$ and $λ_\max$ are defined by \begin{align}λ_\min&=\min\{|λ|:λ\text{ is an eigenvalue of }B\}\\ \text{and }\qquad λ_\max&=\max\{|λ|:λ\text{ is an eigenvalue of }B\} \end{align} Now my question is that whether this relation holds true if $B$ is a negative definite matrix??

Answer 1

Hints:

1. $B$ is negative definite $ \iff -B$ is positive definite.

2. $ \lambda$ is an eigenvalue of $B \iff - \lambda$ is an eigenvalue of $-B.$

Can you proceed ?