Query on Bilinear form

Question

We are given a bilinear form Q(x,y) = x'Ay.

Suppose Q(x,y)>=0 and A is non-singular. So can we then say that x'y>=0 ?

Is there any result/lemma to prove this claim ?

Answer 1

(I hope I understood your question correctly)

Try $$ x=\begin{bmatrix} 1\\ 0\end{bmatrix},\quad y=\begin{bmatrix} -1 \\ 0\end{bmatrix},\quad A=\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}, $$ and you will see that $$ Q(x,y)=x^TAy=1 $$ while $$ x^Ty=-1. $$