Machine learning kernel with complex feature map

Question

I have a question regarding my machine learning lecture where we had to decide whether $$K(x,y)=x_1y_1-x_2y_2$$ is a valid kernel (e.g. for a SVM). My intuition would say that it is a valid kernel since we can display it with: $$\Phi(x)=(x_1, ix_2)\implies K(x,y)=\Phi(x)\Phi(y)$$ with $i$ being the imaginary number. Is that right?

Answer 1

Taking $x = \begin{bmatrix} 0 & 1 \end{bmatrix}^T$, we see that

$$K(x, x) = -1 < 0$$

so that $K$ is not PSD for any set of inputs and therefore is not Mercer.