MATLAB - Colon Notation

MATLAB - The Colon (:) Operator and Indexing

The colon : is one of the most powerful and fundamental operators in MATLAB. It is used for creating vectors, iterating in loops, and, most importantly, for indexing and selecting subsets of data from matrices and arrays. Mastering its use is key to writing efficient, concise, and idiomatic MATLAB code.

Creating Vectors

The colon operator provides a compact syntax for creating sequences of numbers.

• start:end: Creates a vector from start to end with a default increment of 1.
• start:step:end: Creates a vector from start to end with a specified increment of step.

%% Create a vector from 1 to 10 (increment of 1)
v1 = 1:10
% ans = 1  2  3  4  5  6  7  8  9  10

%% Create a vector from 100 down to 50 in steps of -5
v2 = 100:-5:50
% ans = 100  95  90  85  80  75  70  65  60  55  50

%% Create a vector of radian values from 0 to pi
v3 = 0:pi/4:pi
% ans = 0  0.7854  1.5708  2.3562  3.1416

Pro Tip: linspace for Precision

When you need a specific number of points between two values, linspace is often a better choice than the colon operator to avoid floating-point inaccuracies.

% Create exactly 9 points from 0 to pi
v4 = linspace(0, pi, 9)
% ans = 0  0.3927  0.7854  1.1781  1.5708  1.9635  2.3562  2.7489  3.1416

Indexing Arrays and Matrices

This is the most powerful application of the colon operator. It allows for the selection of entire rows, columns, or sub-arrays without writing loops.

Let’s use the following matrix A for our examples:

A = [1 2 3 4; 5 6 7 8; 9 10 11 12]
% A =
%      1     2     3     4
%      5     6     7     8
%      9    10    11    12

Syntax	Purpose	Example	Result
A(i, :)	Selects the entire i-th row.	A(2, :)	[5 6 7 8]
A(:, j)	Selects the entire j-th column.	A(:, 3)	[3; 7; 11]
A(r1:r2, c1:c2)	Selects a sub-matrix from rows r1-r2 and columns c1-c2.	A(2:3, 3:4)	[7 8; 11 12]
A(:, c1:c2)	Selects all rows for a specific range of columns.	A(:, 2:3)	[2 3; 6 7; 10 11]
A(end, :)	The end keyword refers to the last row or column.	A(end, :)	[9 10 11 12]
A(:)	Linear Indexing: Selects all elements and reshapes them into a single column vector (column-major order).	A(1:4)	[1; 5; 9; 2]

Best Practices: Vectorization and Logical Indexing

Vectorization is the practice of using colon notation and other array operators to avoid loops, which is significantly faster and more readable in MATLAB.

% Slow, non-idiomatic way
for i = 1:100
    y(i) = sin(i/10);
end

% Fast, vectorized way
x = 1:100;
y_fast = sin(x/10);

Logical Indexing is a related concept where you use a logical array (containing true and false values) to index into another array. This is extremely powerful for filtering data.

%% Using the matrix A from before
A = [1 2 3 4; 5 6 7 8; 9 10 11 12];

%% Find all elements in A that are greater than 7
large_elements = A(A > 7)
% ans =
%      9
%     10
%      8
%     11
%     12

%% Replace all odd numbers in A with 0
A(mod(A, 2) == 1) = 0
% A =
%      0     2     0     4
%      0     6     0     8
%      0    10     0    12