# Creating a block matrix from arrays of blocks

I am trying to generate a matrix from square blocks. Effectively, I have a $$n×n$$ matrix polynomial $$P(l)$$, the $$qth$$ derivative of $$P(l)$$ with respect to $$l$$, which is denoted by $$P^{(q)} (l)$$, and a block of zeroes, which I’ll just call $$0$$. I have some integer $$k$$ such that if $$k=1$$ then I am generating the matrix

$$R= \begin{pmatrix} P(l) \end{pmatrix}$$

If $$k=2$$ then I should generate

$$R = \begin{pmatrix} P(l) & 0 \\ \frac{1}{1!} P^{(1)}(l) & P(l) \end{pmatrix}$$

If $$k=3$$ then

$$R = \begin{pmatrix} P(l) & 0 & 0 \\ \frac{1}{1!} P^{(1)}(l) & P(l) & 0 \\ \frac{1}{2!} P^{(2)}(l) & \frac{1}{1!} P^{(1)}(l) & P(l) \end{pmatrix}$$

and so forth. Generally,

$$R = \begin{pmatrix} P(l) & 0 & \cdots & 0 & 0 \\ \frac{1}{1!} P^{(1)}(l) & P(l) & \cdots & 0 & 0 \\ \frac{1}{2!} P^{(2)}(l) & \frac{1}{1!} P^{(1)}(l) & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \frac{1}{(k-1)!} P^{(k-1)}(l) & \frac{1}{(k-2)!} P^{(k-2)}(l) & \cdots & \frac{1}{1!} P^{(1)}(l) & P(l) \end{pmatrix}$$

is an $$nk×nk$$ matrix.

I prefer a simple and understandable way and for that I thought to start with a zero matrix $$R$$ of dimensions $$nk×nk$$ and then with two "for" loops to full the initial zero matrix, putting the corresponding derivative which is needed. I’m not sure in what should go as my statement in "for" loops. I found other questions which were similar but more complicated and specific. Any help appreciated, thank you.

• Have you seen ArrayFlatten? – Henrik Schumacher Oct 2 '20 at 9:45

Because of the banded structure of your matrix, you may use the undocumented function SparseArraySparseBlockMatrix for that as follows (using a dummy matrix-valued function P here):

P[i_, l_, n_] := ConstantArray[l, {n, n}];
R[k_, l_, n_] := SparseArraySparseBlockMatrix[
Table[Band[{i, 1}] -> 1/i! P[i, l, n], {i, 1, k}],
{k, k}
]


E.g., R[4, ell, 3] produces a matrix of size {12, 12} with block entries specified by the function P.

• Oh thanks, i didn't know about Band and how to use SparseArraySparseBlockMatrix. – arod Oct 2 '20 at 14:56

First we need same data. I denote your "P(l)" bl p[l] and P^(1)" by der[n,p,x]. Then I define a function "makeblmat" that assembles the block matrix:

m = Table[RandomInteger[{-10, 10}], 2, 2];
p[x_] = Sum[MatrixPower[m, i] x^i, {i, 0, 2}];
derp[n_, p_, x_] := D[p, {x, n}];
makeblmat[k_] :=
Table[If[i >= j, derp[i - j, p[x], x]/! (i - j), 0], {i, k}, {j, k}]//ArrayFlatten;
MatrixForm[makeblmat[2]]


• MatrixForm does not define matrices. Like virtually all built-in symbols ending on Form, it is just a wrapper for display purposes. Better use ArrayFlatten. – Henrik Schumacher Oct 2 '20 at 11:09
• Who said MatrixForm would make a matrix? It is only forto display the output. The matrix is created by "makeblmat" meaning: make block matrix. – Daniel Huber Oct 2 '20 at 11:52
• I pointed out the purpose of MatrixForm because this symbol is frequently an issue on this site. Moreover, makeblmat[k] creates a $k \times k$ matrix of $n\times n$ matrices, but OP asked for a $(nk) \times (nk)$ matrix as result. And MatrixForm does not convert into a $(nk) \times (nk)$ matrix. – Henrik Schumacher Oct 2 '20 at 13:01
• @ Henrik Schumacher I gave the matrix in block form. It is easy to flatten it with ArrayFlatten. – Daniel Huber Oct 2 '20 at 14:14
• That's what I am saying. – Henrik Schumacher Oct 2 '20 at 14:17