eigenvalues of positive definite matrix have both real part and imaginary part positive?

Question

I read about eigenvalues of positive definite matrix have eigenvalues larger than 0, or just stated in books, 'positive', does that mean real and imaginary parts are both positive? Or only the real part matters?

Answer 1

If $x$ is an eigenvector of a positive definite matrix $A$ with eigenvalue $\lambda$, then

$$0 < x^T A x = x^T \lambda x = \lambda |x|^2.$$

Hence $\lambda$ is a positive real number.

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If things are complex replace $x^T$ with $x^*$, the conjugate transpose of $x$, and observe that the same argument works.

Answer 2

Generally, the term "positive definite" is reserved for matrices that are symmetric and thus have real eigenvalues.