# negative semidefinite matrix

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??

1. $$B$$ is negative definite $$\iff -B$$ is positive definite.
2. $$\lambda$$ is an eigenvalue of $$B \iff - \lambda$$ is an eigenvalue of $$-B.$$