I'm working with kernel matrices and I'm stuck with a stupid question I can't answer to.
I have $(z_1, ..., z_m)$ and $(x_1, ..., x_n)$ two sets of points in $\mathcal{X}$. $k$ is a p.d. kernel on $\mathcal{X} \times \mathcal{X}$ and $\mathcal{H}$ is the associated RKHS. Let's say we have $m > n$. I can compute the kernel matrix and its singular value decomposition: $$ [K_{nm}]_{ij} = k(x_i, z_j), ~~K_{nm} = U \Sigma V^\top ~~ \text{and} ~~ \begin{cases} U &\in \mathbb{R}^{n\times n} \\ V &\in \mathbb{R}^{m\times m} \\ \Sigma &= \begin{bmatrix} D & 0 \end{bmatrix} \in \mathbb{R}^{n \times m}. \end{cases} $$
I'm simply wondering if I can relate the spectrum of $K_{nm}$ to the one of $K_m$. Specifically, I can write the eigendecomposition of $K_m$: $$ [K_m]_{ij} = k(z_i, z_j) ~~ \text{and} ~~ K_m = W \Lambda W^\top. $$ My question is the following: is there some basis of $\mathcal{H}$ where the two decompositions coincide? Or do the spectrum of $\Sigma$ and $\Lambda$ have nothing in common?
I tried writing the kernel matrices as a dot product of the embedding, like: $$ K_{nm} = \begin{bmatrix} \phi(x_1)^\top \\ \vdots \\ \phi(x_n)^\top \end{bmatrix} \begin{bmatrix} \phi(z_1) & \cdots & \phi(z_n) \end{bmatrix} $$ but it led me nowhere.
I'm afraid this is the kind of ill-posed questions, let me know if I can provide more details. Any intuition behind these SVD/kernel embeddings stuff would be great. Thanks for your help!