Matrices and Arrays

Open Live Script

MATLAB is an abbreviation for "matrix laboratory." While other programming languages mostly work with numbers one at a time, MATLAB® is designed to operate primarily on whole matrices and arrays.

All MATLAB variables are multidimensional arrays, no matter what type of data. A matrix is a two-dimensional array often used for linear algebra.

Array Creation

To create an array with four elements in a single row, separate the elements with either a comma (,) or a space.

a = [1 2 3 4]

a = 1×4

     1     2     3     4

This type of array is a row vector.

To create a matrix that has multiple rows, separate the rows with semicolons.

a = [1 3 5; 2 4 6; 7 8 10]

a = 3×3

     1     3     5
     2     4     6
     7     8    10

You can also define each row on its own line of code and separate the rows with a newline.

a = [1 3 5
     2 4 6
     7 8 10]

a = 3×3

     1     3     5
     2     4     6
     7     8    10

Another way to create a matrix is to use a function, such as ones, zeros, or rand. For example, create a 5-by-1 column vector of zeros.

z = zeros(5,1)

Matrix and Array Operations

MATLAB allows you to process all of the values in a matrix using a single arithmetic operator or function.

a + 10

ans = 3×3

    11    13    15
    12    14    16
    17    18    20

sin(a)

ans = 3×3

    0.8415    0.1411   -0.9589
    0.9093   -0.7568   -0.2794
    0.6570    0.9894   -0.5440

To transpose a matrix, use a single quote ('):

a'

ans = 3×3

     1     2     7
     3     4     8
     5     6    10

You can perform standard matrix multiplication, which computes the inner products between rows and columns, using the * operator. For example, confirm that a matrix times its inverse returns the identity matrix:

p = a*inv(a)

p = 3×3

    1.0000    0.0000   -0.0000
         0    1.0000   -0.0000
         0    0.0000    1.0000

Notice that p is not a matrix of integer values. MATLAB stores numbers as floating-point values, and arithmetic operations are sensitive to small differences between the actual value and its floating-point representation. You can display more decimal digits using the format command:

format long
p = a*inv(a)

p = 3×3

   0.999999999999996   0.000000000000007  -0.000000000000002
                   0   1.000000000000000  -0.000000000000003
                   0   0.000000000000014   0.999999999999995

Reset the display to the shorter format using

format short

format affects only the display of numbers, not the way MATLAB computes or saves them.

To perform element-wise multiplication rather than matrix multiplication, use the .* operator:

p = a.*a

p = 3×3

     1     9    25
     4    16    36
    49    64   100

The matrix operators for multiplication, division, and power each have a corresponding array operator that operates element-wise. For example, raise each element of a to the third power:

a.^3

ans = 3×3

           1          27         125
           8          64         216
         343         512        1000

Concatenation

Concatenation is the process of joining arrays to make larger ones. In fact, you made your first array by concatenating its individual elements. The pair of square brackets [] is the concatenation operator.

A = [a,a]

A = 3×6

     1     3     5     1     3     5
     2     4     6     2     4     6
     7     8    10     7     8    10

Concatenating arrays next to one another using commas is called horizontal concatenation. Each array must have the same number of rows. Similarly, when the arrays have the same number of columns, you can concatenate vertically using semicolons.

A = [a; a]

A = 6×3

     1     3     5
     2     4     6
     7     8    10
     1     3     5
     2     4     6
     7     8    10

Complex Numbers

Complex numbers have both real and imaginary parts, where the imaginary unit is the square root of -1.

sqrt(-1)

ans = 
0.0000 + 1.0000i

To represent the imaginary part of complex numbers, use either i or j.

c = [3+4i, 4+3j; -i, 10j]

c = 2×2 complex

   3.0000 + 4.0000i   4.0000 + 3.0000i
   0.0000 - 1.0000i   0.0000 +10.0000i

Get Started with MATLAB