I am studying a chapter on positive definites and there is this question in which I have to find whether the quadratic forms are positive definite or not. I have to confirm my answer for this quadratic form: $x^{2}+8xy+y^{2}$. I made it's matrix representation which is $$A = \begin{bmatrix} 1&4\\4&1 \end{bmatrix}$$ The eigenvalues of this matrix are 5 and -3. Since one of the eigenvalue is negative, it must not be a positive definite?
My second doubt is regarding this form: $x^{2}+6xy$. The matrix representation is $$B=\begin{bmatrix} 1&3\\3&0 \end{bmatrix}$$ The eigenvalues of this matrix are $$\lambda = \frac{1 \pm \sqrt{37}}{2}$$ So this quadratic form is also not positive definite, right?