Positive definite matrices: minor criteria vs quadratic form

Question

Let $$A=\begin{bmatrix} 1 & -8\\ 1 & 1 \end{bmatrix}$$

Clearly $\Delta _1>0$ and $\Delta _2 >0$. Hence, we can say that $A$ is positive definite .

But there are $X=\begin{pmatrix} x & y \end{pmatrix}\neq 0 $ such that $X^T \cdot AX <0$

EX : $X=\begin{pmatrix} 1 & 1 \end{pmatrix}$

Does this contradict the fact that $A$ is positive definite according to minor criteria?

Answer 1

Of course, Calculon is right. More precisely, we can give the following definition: $A\in M_n(\mathbb{R})$ is $\geq 0$ IFF for every vector $x$, $x^TAx\geq 0$. This is equivalent to $A+A^T$ is symmetric $\geq 0$.

On the other hand, the test using the principal minors is valid only when the considered matrix is symmetric. Here, $\det(A+A^T)<0$...