TITLE: Does non-symmetric positive definite matrix have positive eigenvalues?
QUESTION [58 upvotes]: I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues.
Does this hold for non-symmetric matrices as well?

REPLY [6 votes]: I am answering the first part of @nukeguy's comment, who asked
``Is the converse true? If all of the eigenvalues of a matrix $$ have positive real parts, does this mean that $^>0$ for any $≠0∈ℝ^$? What if we assume $$ is diagonalizable? –''
I have a counterexample, where $A$ has positive eigenvalues, but it is not positive definite:
$    A=
    \begin{bmatrix}
        7   & -2 & -4  \\
        -17 & 40 & -19 \\
        -21 & -9 & 31
    \end{bmatrix}
$. Eigenvalues of this matrix are  $1.2253$, $27.4483$, and $49.3263$, but it indefinite because if $x_1 = \begin{bmatrix}-48 & -10& -37\end{bmatrix}$ and $x_2 = \begin{bmatrix}-48 &10 -37\end{bmatrix}$, then we have $_1_1^T = -1313$ and $_2_2^T = 37647.$