A block diagonal matrix, also called a diagonal block matrix, is a square diagonal matrix in which the diagonal elements are
square matrices of any size (possibly even 
), and the off-diagonal elements
are 0. A block diagonal matrix is therefore a block matrix
in which the blocks off the diagonal are the zero matrices,
and the diagonal matrices are square.
Block diagonal matrices can be constructed out of submatrices in the Wolfram Language using the following code snippet:
BlockDiagonalMatrix[b : {__?MatrixQ}] :=
Module[{r, c, n = Length[b], i, j},
{r, c} = Transpose[Dimensions /@ b];
ArrayFlatten[
Table[If[i == j,
b[[i]],
ConstantArray[0, {r[[i]], c[[j]]}]],
{i, n}, {j, n}
]
]
]
