Gaussian kernel matrix can be factorized into \((\Phi \textbf{X})^\textbf{H} \Phi \textbf{X} =\textbf{X}^\textbf{H} \Phi^\textbf{H} \Phi \textbf{X} = \textbf{X}^\textbf{H}\textbf{X}\), where \(\Phi\) is Gaussian kernel basis matrix and \(\textbf{X}\) is coefficients matrix of reproducing kernel Hilbert space \(K(\cdot,x) \in \mathcal{H}_K\) https://www.jkangpathology.com/post/reproducing-kernel-hilbert-space/.

A matrix is a system. A system takes input and gives output. A matrix is a linear system. Differentiation and Integration are linear systems. Fourier transformation matches input basis and operator (differentiation) basis. Z transformation matches input digital signal and infinite impulse response filter.

A time-domain sequence can be transformed into a frequency domain by discrete Fourier transformation. The dimension of the discrete Fourier matrix is determined by the length of the time-domain sequence.