How to use ArrayFlatten

Question

I'm trying to use the ArrayFlatten command to create an 8 by 8 matrix of zeros from 4 dimensional column vectors as shown below.

s = ConstantArray[0, 4];
{{s, s, s, s, s, s, s, s}, {s, s, s, s, s, s, s, s}} // MatrixForm
ArrayFlatten[{{s, s, s, s, s, s, s, s}, {s, s, s, s, s, s, s, 
    s}}] // MatrixForm

However, the ArrayFlatten does not do anything here. Please can someone explain what I'm doing wrong and how to fix it? Thanks in advance for any help.

Answer 1

You need s to be a matrix before using ArrayFlatten

Help says

creates a single flattened matrix from a matrix of matrices Subscript[m, ij]

But your s was not a matrix.

s = ConstantArray[0, 4]
InputForm[s]

To make it a matrix of 4 columns you need to do

s = Transpose[{s}]

And now you can use it with ArrayFlatten

s = ConstantArray[0, 4]
s = Transpose[{s}]
(mat = ArrayFlatten[{{s, s, s, s, s, s, s, s}, {s, s, s, s, s, s, s, s}}]) // MatrixForm

Answer 2

@Nasser gave the canonical answer and this is more of a general comment, hopefully useful.

Instead of using ArrayFlatten you can consider ArrayReshape

That is

x = ConstantArray[0, 4];
x = Thread@{x};
matx = ArrayReshape[Table[x, {i, 1, 8}], {8, 8}];

You can check that

MatrixForm@matx

and also

matx === mat

True

with mat being the answer by @Nasser

Answer 3

s = ConstantArray[0, 4];
(t = {s, s, s, s, s, s, s, s}) // MatrixForm

This is a 8x4 matrix, since each s is a row of four elements. Put two of these side by side in a single row, two column arrangement, to get a 8x8 matrix:

ArrayFlatten[{{t, t}}] // MatrixForm

$$\left( \begin{array}{cccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)$$

---

Or start with a block that is a 4x4 matrix:

t = {s, s, s, s}; MatrixForm@t

and make two columns and two rows from these sub-elements.

(ArrayFlatten[{{t, t}, {t, t}}]) // MatrixForm

to get the same 8x8 matrix as above.