Example of a matrix $A$ which has all eigenvalues positive, but is not a general positive definite matrix.

Question

$A_{n\times n}$ is said to be a general positive definite matrix if $x'Ax>0$ for all nonzero $x\in\Bbb{R}^n$.

Obviously for a general positive define matrix every eigenvalue needs to be positive.

I need an example of a matrix $A$ which has all eigenvalues positive, but is not a general positive definite matrix.

Answer 1

An example $\pmatrix{1&-3\cr 0&1}$ in the basis $(e_1,e_2)$, $A(e_1+e_2)=-2e_1+e_2$ $(e_1+e_2)(-2e_1+e_2)^T=-1$.