MATLAB

Basic Matrix Operations

This example shows basic techniques and functions for working with matrices in the MATLAB® language.

First, let's create a simple vector with 9 elements called a.

a = [1 2 3 4 6 4 3 4 5]
a =

     1     2     3     4     6     4     3     4     5

Now let's add 2 to each element of our vector, a, and store the result in a new vector.

Notice how MATLAB requires no special handling of vector or matrix math.

b = a + 2
b =

     3     4     5     6     8     6     5     6     7

Creating graphs in MATLAB is as easy as one command. Let's plot the result of our vector addition with grid lines.

plot(b)
grid on

MATLAB can make other graph types as well, with axis labels.

bar(b)
xlabel('Sample #')
ylabel('Pounds')

MATLAB can use symbols in plots as well. Here is an example using stars to mark the points. MATLAB offers a variety of other symbols and line types.

plot(b,'*')
axis([0 10 0 10])

One area in which MATLAB excels is matrix computation.

Creating a matrix is as easy as making a vector, using semicolons (;) to separate the rows of a matrix.

A = [1 2 0; 2 5 -1; 4 10 -1]
A =

     1     2     0
     2     5    -1
     4    10    -1

We can easily find the transpose of the matrix A.

B = A'
B =

     1     2     4
     2     5    10
     0    -1    -1

Now let's multiply these two matrices together.

Note again that MATLAB doesn't require you to deal with matrices as a collection of numbers. MATLAB knows when you are dealing with matrices and adjusts your calculations accordingly.

C = A * B
C =

     5    12    24
    12    30    59
    24    59   117

Instead of doing a matrix multiply, we can multiply the corresponding elements of two matrices or vectors using the .* operator.

C = A .* B
C =

     1     4     0
     4    25   -10
     0   -10     1

Let's use the matrix A to solve the equation, A*x = b. We do this by using the \ (backslash) operator.

b = [1;3;5]
x = A\b
b =

     1
     3
     5


x =

     1
     0
    -1

Now we can show that A*x is equal to b.

r = A*x - b
r =

     0
     0
     0

MATLAB has functions for nearly every type of common matrix calculation.

There are functions to obtain eigenvalues ...

eig(A)
ans =

    3.7321
    0.2679
    1.0000

... as well as the singular values.

svd(A)
ans =

   12.3171
    0.5149
    0.1577

The "poly" function generates a vector containing the coefficients of the characteristic polynomial.

The characteristic polynomial of a matrix A is

$$det(\lambda I - A)$$

p = round(poly(A))
p =

     1    -5     5    -1

We can easily find the roots of a polynomial using the roots function.

These are actually the eigenvalues of the original matrix.

roots(p)
ans =

    3.7321
    1.0000
    0.2679

MATLAB has many applications beyond just matrix computation.

To convolve two vectors ...

q = conv(p,p)
q =

     1   -10    35   -52    35   -10     1

... or convolve again and plot the result.

r = conv(p,q)
plot(r);
r =

     1   -15    90  -278   480  -480   278   -90    15    -1

At any time, we can get a listing of the variables we have stored in memory using the who or whos command.

whos
  Name      Size            Bytes  Class     Attributes

  A         3x3                72  double              
  B         3x3                72  double              
  C         3x3                72  double              
  a         1x9                72  double              
  ans       3x1                24  double              
  b         3x1                24  double              
  p         1x4                32  double              
  q         1x7                56  double              
  r         1x10               80  double              
  x         3x1                24  double              

You can get the value of a particular variable by typing its name.

A
A =

     1     2     0
     2     5    -1
     4    10    -1

You can have more than one statement on a single line by separating each statement with commas or semicolons.

If you don't assign a variable to store the result of an operation, the result is stored in a temporary variable called ans.

sqrt(-1)
ans =

   0.0000 + 1.0000i

As you can see, MATLAB easily deals with complex numbers in its calculations.