what represents this strange matrix

Question

But I just encountered a very strange type of matrix.

What is the name of this type of matrix? How can it be solved? The final result should be a $4\times 4$ matrix, but I only see $5$ elements on very strange places in this matrix.

Answer 1

The matrix $K$ is written as a block matrix, which means that the four entries you see are actually other matrices (of differing dimensions).

So $\sigma$ is just a number (in whatever field you're working over), i.e., a $1\times 1$ matrix. $Z$ is the $3\times 1$ matrix listed there and $Z^T$ is the transpose: a $1\times 3$ matrix. This leaves $S-\sigma I = B+B^T-\sigma I$. From the context (or reading the paper), we can infer that $B$ must be a $3\times 3$ matrix and $I$ is the $3\times 3$ identity matrix.

Something like $$K=\begin{bmatrix} \color{red}{s_{1,1}-\sigma} &\color{red}{s_{1,2}} & \color{red}{s_{1,3}} & \color{blue}{z_1}\\ \color{red}{s_{2,1}} & \color{red}{s_{2,2}-\sigma} & \color{red}{s_{2,3}} & \color{blue}{z_2}\\ \color{red}{s_{3,1}} & \color{red}{s_{3,2}} & \color{red}{s_{3,3}-\sigma} & \color{blue}{z_3}\\ \color{blue}{z_1} & \color{blue}{z_2} & \color{blue}{z_3} & \sigma\end{bmatrix}$$