Positive Matrices and Linear Forms

Question

Fix a vector $\vec{b}$ and a positive definite (but not necessarily symmetric) matrix $A$, can we prove that the fraction $$ \frac{\vec{b}^TA\vec{y}}{\vec{b}^T\vec{y}} $$ always has the same sign (If $\vec{b}^T\vec{y}\neq 0$)?

Answer 1

No. Consider $A = \begin{pmatrix}\frac{1}{2} & 0 \\ 0 & 2\end{pmatrix}$, $b = (1,1)^T$, and $y = (2,-1)^T$. $b^Ty = 1>0$, but $b^TAy = b^T(1,-2)^T = -1<0$. Therefore the fraction is negative for this choice of $y$, but we find that the fraction is positive for $y=(1,0)^T$