Multidimensional Matrices

There is no limit to the number of dimensions, apart from the maximum 32-bit integer size and the machine memory required to store the matrix.

b = ones(2);  ==>  b is a 2x2 matrix

b(:,:,3) = [5,6;7,8] ==> b is now a 2x2x3 matrix as follows:

slice(:, :, 1) = 
[Matrix] 2 x 2
1  1
1  1
slice(:, :, 2) = 
[Matrix] 2 x 2
0  0
0  0
slice(:, :, 3) = 
[Matrix] 2 x 2
5  6
7  8

For example, running this script:

m3m = ones(2,2);
m3m(:,:,2) = [5,6;7,8];
m3m(:)
m3m(:,:,:)
m3m(1)
m3m(:,1,:)
m3m(1,:,2)

produces this output:

ans = [Matrix] 8 x 1
1
1
1
1
5
7
6
8
ans = 
slice(:, :, 1) = 
[Matrix] 2 x 2
1  1
1  1

slice(:, :, 2) = 
[Matrix] 2 x 2
5  6
7  8

ans = 1
ans = 
slice(:, :, 1) = 
[Matrix] 2 x 1
1
1

slice(:, :, 2) = 
[Matrix] 2 x 1
5
7

ans = [Matrix] 1 x 2
5  6

The colon operator provides considerable flexibility in how data can be assigned to a matrix. Consider this example:

m(2,:,:) = [1,2;3,4]

The left-hand side is expecting to assign data in dimensions 2 and 3, but the right-hand side has data in dimensions 1 and 2. Strictly, this might be regarded as an error. However, it is useful to allow an implicit assignment of the two right-hand side dimensions to the two colon dimensions on the left to produce the following result:

m =
slice(:, :, 1) =
[Matrix] 2 x 2
0  0
1  3

slice(:, :, 2) =
[Matrix] 2 x 2
0  0
2  4

By allowing both m(:,:,2) and m(2,:,:) on the left-hand side, you can interpret the dimensions of m as either (row, col, page) or (page, row, col). This kind of flexibility applies to all numbers of dimensions and colons.

Similarly, to 2D matrices, setting one value of an element defines the size. For example:

m3m(2,2,2) = 1
m3m = 
slice(:, :, 1) = 
[Matrix] 2 x 2
0  0
0  0

slice(:, :, 2) = 
[Matrix] 2 x 2
0  0
0  1

Many functions do not operate on multidimensional matrices (for example, matrix inversion). However, it can often be useful to extract data two dimensions at a time and reformat them into a two-dimensional matrix using the squeeze or reshape functions. Consider this sequence that creates a two-dimensional matrix b from dimensions 1 and 3 of m.

m(2,:,:) = [1,2;3,4];
b = m(:,1,:)  ==> here b is a 2x1x2 matrix
b = squeeze(b)  ==> here b is now a 2x2 matrix

A matrix can also be reduced to a 2D matrix by deleting higher dimensions. For example:

b = ones(2,2)
b(:,:,2) = [5,6;7,8];  ==> here b is a 2x2x2 matrix
b(:,:,2) = [];  ==> here b is now a 2x2 matrix