Prove that ${f(x,y) = x^Txx^Tyy^Ty}$ is a valid kernel The rules I can use without a proof are the following ones:

The thing that I know that a kernel is valid if it is symmetric, e.g. ${f(x,y) = f(y,x)}$ and if ${c^TKc >= 0}$, for each vector ${c \neq 0}$, but I don't know how to prove it, and don't know how to get the matrix K (if this is the correct way to solve this exercise).

Let $\phi(\mathbf x)=\mathbf x\mathbf x^T\mathbf x$ be a nonlinear transformation for $\mathbf x$. Then, you can use the fourth rule in your screenshot, because $k(\mathbf x, \mathbf  y)=\phi(\mathbf x)^T\phi(\mathbf y)$ is a valid kernel since $k_3(\mathbf x,\mathbf y)=\mathbf x^T\mathbf y$ is a valid kernel.
