# ArrayFlatten

ArrayFlatten[{{m11,m12,},{m21,m22,},}]

creates a single flattened matrix from a matrix of matrices mi j.

ArrayFlatten[a,r]

flattens out r pairs of levels in the array a.

# Details • ArrayFlatten requires that the blocks it flattens have dimensions that fit together.
• ArrayFlatten can be used to form block matrices from arrays of blocks.
• For a matrix of matrices, ArrayFlatten[a] yields a matrix whose elements are in the same order as in MatrixForm[a].
• ArrayFlatten[a] is normally equivalent to Flatten[a,{{1,3},{2,4}}]. »
• ArrayFlatten[a,r] is normally equivalent to Flatten[a,{{1,r+1},{2,r+2},,{r,2r}}].
• For a tensor with rank 2r, ArrayFlatten[a,r] gives a tensor with rank r.
• In ArrayFlatten[{{m11,m12,},{m21,m22,},}], all the matrices mi j in the same row must have the same first dimension, and matrices mi j in the same column must have the same second dimension.
• In general, in ArrayFlatten[a,r], all the k dimensions of a[[i1,i2,, ]] must be equal for each possible value of ik .
• Elements at level r whose array depth is less than r are treated as scalars, and are replicated to fill out a rank-r array of the appropriate dimensions.
• ArrayFlatten works with SparseArray objects. »

# Examples

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## Basic Examples(2)

Create a block matrix by flattening out a matrix of matrices:

Use 0s to represent zero matrices:

## Scope(4)

Flatten a rank-4 array to rank 2:

Flatten only the first four levels of a rank-6 array:

Flatten a rank 6 array to rank 3:

ArrayFlatten works with SparseArray objects:

Make a sparse matrix from a block matrix of SparseArray objects:

## Applications(5)

Put together many copies of a "tile":

Iterate a 2D substitution system:

Iterate a 3D substitution system:

Form an involutory matrix , such that :

Check that the resulting matrix is involutory:

Form a block matrix semidiscretization of the wave equation with n spatial points:

Differentiation matrix for second-order approximation of with periodic boundary conditions:

Identity matrix of size n:

Form block matrix a for system where :

Identity matrix of size 2 n:

Set up an initial condition vector for :

Approximate the solution at using the backward Euler method with time step k:

Show and at :

## Properties & Relations(3)

MatrixForm displays matrices of matrices in the same order as ArrayFlatten:

ArrayFlatten is a special case of Flatten:

KroneckerProduct is defined as ArrayFlatten of an Outer product: