Let $\phi = [x1^2, x1x2, x2^2]^\top$.
$W_1 ^\top \phi(x)$ and $W^{* \top} \phi(x)$ are positive definite functions and $A$ is a rank-1 positive semi definite matrix ($0<$one eigenvalue$<1$, the rest of eigenvalues are $0$).
If we update $W$ as follows: $W_2^\top \phi(x) =W_1 ^\top (I-e_{lr}A) ^\top\phi(x) + e_{lr}W^{* \top}A ^\top \phi(x)$, is $W_2^\top \phi(x)$ a positive definite function?
I think if $e_{lr}$ is sufficiently small, it is true.