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A is real matrix from order nXn. We know that A gives : $<Av,v>=0$ for v vector in $R^n$.

So what must exist?

  1. Every eigenvalue of A is real.
  2. A is not Invertible
  3. A is Hermitian
  4. A is not Hermitian
  5. all answers are incorrect

I think 2 is the right answer ( can be more than 1 answer that are true) but im not sure

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1 Answer 1

up vote 1 down vote accepted

$\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ is a counter-example to 1 and 3. The zero matrix is a counter example to 4.

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This is also a counter-example to 2. So 5 is the correct answer. –  1015 Feb 4 '13 at 0:16
    
Sure, the identity matrix is a counter example to 4. –  Damien L Feb 4 '13 at 7:53
    
No, the identity matrix does not fulfill the condition $<Av,v>=0$ for all $v$. –  1015 Feb 4 '13 at 13:09

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