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Are eigenvalues of $A$ all with positive real parts if and only if $x^TAx>0$ for any $x$? $A$ is non symmetric. If this is true, if $B=-B^T$, then if the eigenvalue of $A$ are with positive real parts, so will be the eigenvalue of $(A+B)$.

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If $x^TAx$ is positive, then the eigenvalues are positive (irrespective of the matrix being symmetric or not), but the converse is not true for non-symmetric matrices. (The converse is true for symmetric matrices.)

For instance, consider the Jordan matrix $$A = \begin{bmatrix}1 & -4\\0 &1 \end{bmatrix}$$ The eigenvalues are clearly positive (since the eigenvalues are both $1$) but the matrix is not positive definite.

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