Create block matrix with secondary diagonals using Band function

Question

Given matrices m1, ... , mk and matrices n1, ... , nk-1, I would like to define a block matrix $$ m=\left( \begin{array}{rrrr}m1 & n1 & 0 & 0 \\0 & m2 & \ddots & 0 \\\vdots & \ddots & \ddots & nk-1 \\0 & \dots & 0 & mk \\\end{array}\right) $$ with blocks mi on the diagonal and blocks nj on the secondary diagonal. The sizes of the matrices are compatible.

I would like to define m as an sparse array using the Band function, but I didn't manage. Is it possible?

Edit: Before, I called the matrices m[i] and n[j] which was misleading, see kglr's answer

Answer 1

Update:

ClearAll[sA]
sA = SparseArray[{Band[{1, 1}] -> #, Band[{1, 1 + Last@Dimensions[#[[1]]]}] -> #2}] &;

Example:

SeedRandom[1]
k = 4;
{rowdims, coldims} = RandomInteger[{2, 4}, {2, k}];

ClearAll[m1, m2, m3, m4, n1, n2, n3, ma, mb, mc, md, na, nb, nc];

ms = {m1, m2, m3, m4} = MapThread[Array[Function[{x, y}, Subscript[#, x, y]], #2] &, 
   {{ma, mb, mc, md}, Transpose[{rowdims, coldims}]}];

ns = {n1, n2, n3} = MapThread[Array[Function[{x, y}, Subscript[#, x, y]], #2] &, 
   {{na, nb, nc}, Transpose[{Most@rowdims, Rest@coldims}]}];

sA[ms, ns] // MatrixForm

sA[Map[Style[#, Blue] &, ms, {-2}], Map[Style[#, Red] &, ns, {-2}]] // MatrixForm

Original answer:

sa[k_Integer] := SparseArray[{Band[{1, 1}] -> Array[m, k], 
    Band[{1, 2}] -> Array[n, k - 1]}, {k, k}];

sa[5] // MatrixForm // TeXForm

$\left( \begin{array}{ccccc} m(1) & n(1) & 0 & 0 & 0 \\ 0 & m(2) & n(2) & 0 & 0 \\ 0 & 0 & m(3) & n(3) & 0 \\ 0 & 0 & 0 & m(4) & n(4) \\ 0 & 0 & 0 & 0 & m(5) \\ \end{array} \right)$