Let $x = ab$ where, x is a nonlinear function of $a,b$. I want to linearize it in the form of $x = Fy$, where, $y = [a \ \ b]^T$. If I use the Jacobian of $x$ w.r.t. $ a$ and $b$ as $F$, then I get $x = [b \ \ a]\times y$. Replacing the value of $y$, we get, $x = [b \ \ a]\times [a\ \ b]^T = 2ab$. But $ab = x$, so I get $x = 2x$. What is wrong here. I know that if we linearize a nonlinear function $Y = F(X)$ in the form of $Y = AX$, using Taylor series expansion, then $AX$ is not necessarily equal to $F(X)$ because of the omission of higher order terms, but in the above case does it make sense to get $x = 2x$.
Is it correct to write $x = [0.5b \ \ 0.5a]\times[a\ \ b]^T$ and use $F = [0.5b \ \ 0.5a]$ as Jacobian?