0
$\begingroup$

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??

$\endgroup$
0
$\begingroup$

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 ?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.