MATLAB - Compatible Arrays

MATLAB - Implicit Expansion

Implicit expansion, sometimes known as broadcasting, is a powerful MATLAB feature that simplifies vector and matrix operations. When you perform element-wise operations on arrays of different but compatible sizes, MATLAB automatically expands the smaller dimensions to match the larger ones, avoiding the need for manual replication of arrays.

This feature was formally expanded and named in MATLAB R2016b. Two arrays are compatible for an operation if, for every dimension, the dimension sizes are either equal or one of them is 1.

Core Principles of Implicit Expansion

• Scalar Expansion: The most common case. A scalar can be combined with any size array. The scalar is effectively expanded to match the size of the other array.
• Vector and Matrix Expansion: A row or column vector can be combined with a 2D matrix if its length matches the corresponding matrix dimension (row vector length matches number of columns, column vector length matches number of rows).

Example 1: Scalar Expansion

Adding a scalar to a matrix. MATLAB expands the scalar to a matrix of the same size and then performs the addition.

matrix = [1, 2; 3, 4];
scalar = 10;

% Implicit expansion of 'scalar' to a 2x2 matrix
result = matrix + scalar;

disp('Original Matrix:');
disp(matrix);
disp('Result after adding a scalar:');
disp(result);

The output demonstrates the element-wise addition:

Original Matrix:
     1     2
     3     4

Result after adding a scalar:
    11    12
    13    14

Example 2: Vector and Matrix Expansion

Subtracting a column vector from a matrix. This is a common operation for normalizing data, such as subtracting the mean of each row.

% A 3x4 matrix of sample data
data = [10, 20, 30, 40; ...
        15, 25, 35, 45; ...
         5, 10, 15, 20];

% A 3x1 column vector of row means
rowMeans = mean(data, 2); % The '2' indicates mean across columns (i.e., for each row)

% Subtract the mean from each element in its corresponding row.
% The 3x1 'rowMeans' vector is implicitly expanded to a 3x4 matrix.
centeredData = data - rowMeans;

disp('Original Data:');
disp(data);
disp('Row Means (3x1 vector):');
disp(rowMeans);
disp('Centered Data (after implicit expansion):');
disp(centeredData);

The output shows how the column vector was subtracted from each column of the matrix:

Original Data:
    10    20    30    40
    15    25    35    45
     5    10    15    20

Row Means (3x1 vector):
   25.0000
   30.0000
   12.5000

Centered Data (after implicit expansion):
  -15.0000   -5.0000    5.0000   15.0000
  -15.0000   -5.0000    5.0000   15.0000
   -7.5000   -2.5000    2.5000    7.5000

Example 3: Row and Column Vector Expansion

When you combine a column vector (size M-by-1) and a row vector (size 1-by-N), the result is an M-by-N matrix. This is a powerful way to generate grids or tables of values.

colVector = [10; 20; 30];  % 3x1 vector
rowVector = [1, 2, 3, 4]; % 1x4 vector

% The column vector is expanded across 4 columns
% The row vector is expanded down 3 rows
result = colVector + rowVector;

disp('Result of adding a 3x1 and 1x4 vector:');
disp(result);

The resulting 3x4 matrix is:

Result of adding a 3x1 and 1x4 vector:
    11    12    13    14
    21    22    23    24
    31    32    33    34

Common Errors: Dimension Mismatch

An error occurs if array dimensions are not compatible. For an operation to work, each dimension must either be the same size, or one of them must be 1.

% Incompatible arrays
A = [1, 2, 3];   % size 1x3
B = [4; 5];      % size 2x1

try
    C = A + B; % This will fail
catch ME
    disp('Error caught as expected:');
    disp(ME.message);
end

This code will produce an error because the dimensions (1x3 and 2x1) are not compatible:

Error caught as expected:
Error using  +
Matrix dimensions must agree.