Tutorial · Beginner · 10 min

Vectors, matrices and indexing

Everything in MathLab is a matrix. Learn to build them, combine them and pick out exactly the elements you need.

1. Rows, columns and ranges

   Spaces or commas separate elements; semicolons start a new row. a:b counts in steps of 1, a:step:b in other steps.

   row = [1 2 3]
   col = [1; 2; 3]
   r = 0:0.25:1

   Try it ↗

   row = 1×3
     1  2  3
   col = 3×1
     1
     2
     3
   r = 1×5
        0  0.25   0.5  0.75     1

2. Matrices and their size

   A matrix is rows separated by semicolons. size gives rows × columns, and A' is the transpose.

   A = [1 2 3; 4 5 6]
   size(A)
   A'

   Try it ↗

   A = 2×3
     1  2  3
     4  5  6
   ans = 1×2
     2  3
   ans = 3×2
     1  4
     2  5
     3  6

3. Element-wise or matrix maths?

   * is matrix multiplication; .*, ./ and .^ work element by element. Most plotting code needs the dotted versions: x.^2, not x^2.

   x = [1 2 3];
   x .* x            % element-wise: [1 4 9]
   x * x'            % matrix product: 1 + 4 + 9 = 14
   x.^2

   Try it ↗

   ans = 1×3
     1  4  9
   ans = 14
   ans = 1×3
     1  4  9

   Tip: Getting "Inner matrix dimensions must agree"? You probably meant .* instead of *.

4. Picking elements

   Indices start at 1. end means the last one, : means all. A condition like x > 0 gives a mask that selects matching elements.

   A = magic(4)
   A(2, 3)           % row 2, column 3
   A(:, 1)'          % the first column, as a row
   A(end, :)         % the last row
   v = -3:3;
   v(v > 0)          % only the positive values

   Try it ↗

   A = 4×4
     16   2   3  13
      5  11  10   8
      9   7   6  12
      4  14  15   1
   ans = 10
   ans = 1×4
     16   5   9   4
   ans = 1×4
      4  14  15   1
   ans = 1×3
     1  2  3

5. Solving a linear system

   To solve A x = b, use A \ b. It is faster and more accurate than computing inv(A).

   A = [4 -2 1; -2 4 -2; 1 -2 4];
   b = [11; -16; 17];
   x = A \ b
   residual = norm(A*x - b)

   Try it ↗

   x = 3×1
      1
     -2
      3
   residual = 0