Let $A\in\Bbb R^{n\times n} $be a symmetric matrix.

$1)$ Prove that its eigenvalues are real and that any two eigenvector associated with different eigenvalues are orthogonal.

$2)$ Is this true if $A\in \Bbb C ^{n\times n} $ and $A$ is hermitian?

I can solve $1)$

For $2)$ I don't see why the conclusions wouldn't hold: if it's hermitian its eigenvalues must be real and if it's hermitian the it's normal so the eigenvector for different eigenvalues will be orthogonal. However I feel this maybe a trick question. Is there anything I'm missing?

  • $\begingroup$ This is known as the spectral theorem. No trick. $\endgroup$ Aug 16, 2014 at 22:43
  • $\begingroup$ I think you're absolutely correct. Doesn't look like a trick question to me. (Isn't the point of this question to prove what you stated?) $\endgroup$
    – Tunococ
    Aug 16, 2014 at 23:32
  • $\begingroup$ @milo : You don't need to write $nxn$; you can write $n\times n$. I edited accordingly. $\endgroup$ Aug 17, 2014 at 0:21
  • $\begingroup$ Try to imitate this proof. $\endgroup$ Aug 17, 2014 at 2:26


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