Question

Is it possible for a symmetric matrix to have a negative eigenvalues? Or all symmetric matrices will ALWAYS have positive eigenvalues?

Answer 1

Err, a diagonal matrix is symmetric, simply place a negative value in one of the entries of the diagonal matrix and you have a symmetric matrix with negative eigenvalues.

Answer 2

\[
\left[ \begin {array}{cccc} -1&0&0&0\\0&-2&0&0
\\ 0&0&-3&0\\0&0&0&-4\end {array}
\right]
\]

Answer 3

So does this imply that other than diagonal matrices, for normal non-diagonal symmetric matrices, there are still chances that their eigenvalues turn out to be negative too?

Answer 4

Yes, symmetric matrices can have negative eigenvalues.

Answer 5

ahh...I see...Thanks a lot! :)