I'm doing some calculations with diagonal block matrices and I'd like to write the components of the block matrix explicitly with respect to the blocks. For diagonal matrices this is easy using the Kronecker delta. For instance, given the matrix:
$ A = \left(\begin{matrix} a_1 &0&0\\ 0&a_2&0\\ 0&0&a_3 \end{matrix}\right)$
the corresponding components are (no summation over repeated indices):
$ A_{ij} = a_i\delta_{ij} $
Is there a similar way to write the components of a general block diagonal matrix? What I mean by "similar" is that the expression should use the components of the blocks and Kronecker deltas. I'd refer to the components of the blocks as $A_{ij}^{(k)}$, where $k=1,\ldots,n$, $i,j=1,\ldots m_k$, with $n$ the number of blocks and $m_k$ (squared) the dimension of the $k$-th block.