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If eigenvalues of a symmetric matrix are positive, does that mean it is positive definite? please give an example of why this is incorrect.

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  • $\begingroup$ What are you using as the definition of "positive definite"? Are you considering complex matrices that are symmetric but not hermitian? $\endgroup$ – Robert Israel Feb 18 '16 at 19:18
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The symmetric (but not hermitian) matrix $$ \pmatrix{2 & i\cr i & 0\cr}$$ has $1$ as its only eigenvalue, but it is not positive definite.

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  • $\begingroup$ can you also give an example with integer numbers and not complex? $\endgroup$ – mehrdad Feb 18 '16 at 19:41
  • $\begingroup$ No, because the statement is true for real symmetric matrices. $\endgroup$ – Robert Israel Feb 18 '16 at 21:48

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