Meaning of matrix 'diag' operator with matrix arguments

Question

I came across this definition in a paper and can't figure out what it is supposed to represent:

I understand that there is a 'diag' operator which when given a vector argument creates a matrix with the vector values along the diagonal, but I can't understand how such an operator would work on a set of matrices.

Answer 1

$\operatorname{diag}$ with matrix arguments creates a block-diagonal matrix. If, say, $A$ is an $M$ by $M$ matrix, and $B$ is $N$ by $N$, then

$$\operatorname{diag}(A,B) = \begin{pmatrix} A & 0_{M,N} \\ 0_{N,M} & B\end{pmatrix},$$

where $0_{M,N}$ is the $M$ by $N$ matrix of zeros.