Removing the $i$-th row and column from the kernel matrix

In the sliding-window approach from [12], the first row and column of the upsized kernel matrix $\breve{{\mathbf K}}_n$ are removed in every iteration, yielding the ``downsized'' matrix ${\mathbf K}_n$. The inverse of this matrix can be obtained efficiently based on the knowledge of $\breve{{\mathbf K}}_n^{-1}$, as follows

$$\breve{{\mathbf K}}_n = \begin{bmatrix}a & \textbf{b}^T \textbf... ...matrix} \Rightarrow {\mathbf K}_n^{-1} = \textbf{G} - \textbf{f}\textbf{f}^T/e.$$ (6)

The proposed method requires to remove an arbitrary row and column of the matrix $\breve{{\mathbf K}}_n$. In order to do this, the matrix inversion formula can be extended by applying a few permutations, based on

$${\mathbf P}_i = \begin{bmatrix}0 & {\mathbf 0} & 1 & {\mathbf 0} ... ...0} & {\mathbf 0} {\mathbf 0} & {\mathbf 0} & {\mathbf I}_{M-i} \end{bmatrix},$$ (7)

in which ${\mathbf I}_{j}$ is the unit matrix of size $j$ and ${\mathbf 0}$ is the all-zeroes matrix of adequate dimensions. Notice that ${\mathbf P}_i^{-1} = {\mathbf P}_i$ and ${\mathbf Q}_i^{-1} = {\mathbf Q}_i^T$. Algorithm 1 summarizes the steps necessary to obtain the required inverse matrix. In the first step, the result of pre-and post-multiplying by ${\mathbf P}_i$ is an exchange of the first and $i$-th row and column. In the last step, pre- and post-multiplying a matrix by ${\mathbf Q}_i$ puts its $i$-th row and column in front of the others. In practice, these calculations can be implemented as fast matrix operations.