Stage 1
MATLAB Self-do-tutorial (Part 1)
Contents
1.1 Introduction
1.2 Matrices
1.3 Matrix operations and functions
1.4 Graphics
1.1 Introduction
MATLAB is a technical software environment based on matrix manipulations offering both numerical
processing and visualization tools. MATLAB combines numerical analysis, matrix-computation,
signal processing and visualization tools with a simple user-friendly environment. Problems and
solutions are expressed using statements, which resemble standard mathematical expressions without
the need for developing software using traditional programming techniques such as C, Pascal or
Fortran.
The goal of the self-do-tutorial is offering the student with some basic MATLAB knowledge
enough for a successful completion of the practices of this course. This means that simplicity has
priority over fast execution. Fantasy over creativity.
Attainment targets
The goals of the Self-do-tutorial are to be able:
to generate and manipulate matrices with MATLAB,
to visualize results in MATLAB,
to write MATLAB m-files.
Organization of this chapter
The MATLAB introduction is divided into five subsections. Matrices and matrix-functions are
discussed in the sections 1.2 and 1.3. A detailed description of MATLAB’s graphical visualization
tools is given in Section 1.4
1.2 Matrices
MATLAB works with a single object, an matrix, possibly including complex elements. Scalars are
treated as matrices, column vectors as matrices and row vectors as matrices. In this section, we shall
describe in details how to work with matrices, how to write/read variables and how to select
submatrices.
__
___
___
Generating matrices
To declare the (row) vector with, for example, elements 1, 3 and 4, we can type the following:
_
>> a = [1 3 4]
a =
1 3 4
Another way to declare vectors is to use an arbitrary increment between its elements, for example:
>> b = [0:2:6]
b =
1
MATLAB
0 2 4 6
or simply:
>> b = 0:2:6
b =
0 2 4 6
This results in an integer vector with an increment of 2 between successive elements. The default
increment size (if not explicitly given) is 1. The transpose of a matrix or a vector is given by the
operation ’, that is,
>> a’
ans =
1
3
4
results in the transposed of the vector . Special attention should be given to the fact that the MATLAB
operation ’ transposes and complex-conjugates matrices. For transposition only, use the command
.’. The last result of any operation that is not saved in a specific variable is kept by MATLAB in the
special variable ans.
It is possible to generate a column vector by separating the different rows using the semicolon,
i.e.,
_
>> [1; -3; 4]
ans =
1
-3
4
Of course we can also define vectors with complex elements, using either or as the imaginary unit. For
example:
_
_
>> c = [2+i; -i]
c =
2.0000 + 1.0000i
0 - 1.0000i
but also
>> c = [2-j j]’
c =
2.0000 + 1.0000i
0 - 1.0000i
Note that the ’ operation not only transposes the vector, but also complex conjugates it. Additionally,
one can define a new imaginary unit, for example im unit = sqrt(-1).
A matrix is defined in the same way as a vector. This is done by defining either a row vector of
columns, or a column consisting of rows, where the different rows are separated by a semicolon, e.g.,
>> A = [[1 4 7]’ [2 5 8]’ [3 6 9]’]
A =
1 2 3
4 5 6
7 8 9
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or
>> A = [1
A =
1 2
4 5
7 8
2 3; 4 5 6; 7 8 9]
3
6
9
but also
>> A = [[1;4;7] [2;5;8] [3;6;9]];
Note that MATLAB is case sensitive (a A) and that terminating the command line with a semicolon
suppresses the display of its result. This feature will become essential when we want to display the
final result of complex computations, not being interested in intermediate results. MATLAB includes
many built-in functions for automatic matrix creation. Among these are:
zeros (matrix with zeros),
ones (matrix with ones),
eye (identity matrix).
__
Submatrices
MATLAB allows the selection of specific parts of a vector. For instance, to view only the first
three elements of the vector , we can type:
_
>> b(1:3)
ans =
0 2 4
To view the first and third element, type:
>> b([1 3])
ans =
0 4
If we wish to select all the elements beginning at a given position till (and including) the last
element, we can use the variable end:
>> b(2:end)
ans =
2 4 6
Note that the first vector element in MATLAB is indexed 1, and not 0. So,
>> b(0)
??? Index exceeds matrix dimensions.
This is one of the most common error messages an unexperienced MATLAB user will encounter. Just
like vectors, we can select matrix elements, for example the element (1,2) of matrix :
_
>> A(1,2)
ans =
2
or the first two columns
>> A(1:3,1:2)
ans =
1 2
3
MATLAB
4 5
7 8
The same result can be achieved by typing:
>> A(:,1:2)
ans =
1 2
4 5
7 8
Here, the colon means “all elements”. To select the entire matrix, we thus type:
>> A(:,:)
A =
1 2 3
4 5 6
7 8 9
Variables
We can get the list of all currently defined variables using the commands who and whos:
>> who
Your variables are:
A
a
ans
b
>> whos
Name Size
A
3x3
a
1x3
ans
3x2
b
1x4
c
2x1
Grand total is 24
c
Bytes
Class
72
double
24
double
48
double
32
double
32
double
elements using 208
array
array
array
array
array (complex)
bytes
To view the dimension of a specific variable, use the size command, for example:
>> size(b)
ans =
1 4
It is often sufficient to know only the length of a vector. To do so, use the command length:
>> length(b)
ans =
4
The clear command clears variables:
>> clear c
>> who
Your variables are:
A
a
ans
b
When exiting a MATLAB session, all variables are destroyed. However, with the command save
it is possible to save all declared variables to the file “matlab.mat” before exiting the session. Later,
when starting a new session, one can read these saved variables by typing load. It’s also possible to
save only specific variables. For example to save the matrix to a file named “A matrix.mat” type:
_
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>> save A_matrix A
This file can be read later with the command:
>> load A_matrix
If you wish to write data in an ASCII format, use the option -ascii. Use the help function to view
other available options of save (help save).
We can format an output file with the use of the commands fopen, fprintf and fclose
(see the help information of these commands). Writing the vector to a file “a vector.data” using 6
digit notation of which three following the decimal point, we can type:
_
>> fp = fopen(’a_vector.data’, ’w’);
>> fprintf(fp, ’%6.3f %6.3f %6.3f n’, a(1), a(2), a(3));
>> fclose(fp);
_
If fp = 1 (default value), the output will be sent to the standard output (the monitor).
1.3 Matrix operations and functions
In this section, we explore different matrix manipulation techniques.
Matrix operations
As you might have noticed by now, MATLAB uses the standard linear-algebra notation. Both linear
and non-linear operations, like addition, subtraction, multiplying, and raising to a power (+,-,* and
^, respectively) can be easily achieved. Pay attention to matrix and vector dimensions.
See the following examples:
>> A*a
??? Error using ==> * Inner matrix dimensions must agree.
>> A*a’
ans =
19 43 67
>> b(2:4)*A
ans =
60 72 84
>> a + b([1 2 4])
ans =
1 5 10
It is sometimes desirable to perform element-wise operations. This is done by placing a point before
the mathematical symbol. For example, to raise each element of matrix to the power of two, we type:
_
>> A.ˆ2
ans =
1 4 9
16 25 36
49 64 81
This is not equivalent to
>> Aˆ2
ans =
30
36
42
5
MATLAB
66 81 96
102 126 150
indicating the matrix product
Matrix functions
MATLAB has many built-in functions. Some (e.g. sin) work on scalars. When called with a matrix
argument, those functions will work on each element separately:
>> sin(b)
ans =
0
0.9093
>> max(sin(b))
ans =
0.9093
-0.7568
-0.2794
In perspective of random variables and stochastic processes functions will be often work on random
outcomes of experiments. On generating random numbers simulating outcomes and on analyzing
outcomes of experiments. In the following we use the function rand,which generates from a pseudo
random number generator a number in the interval (0,1). Each time you use rand your specific
answer will be different from ours but must have the same properties:
>> rand
ans =
0.0153
>> rand
ans =
0.7468
>> rand
ans =
0.4451
Till now the outcome is shown as the ans. If the value of rand e.g. is compared to a scalar:
>> rand>0.5
ans =
1
>> rand>0.5
ans =
0
only the result is presented and we can’t check the quality of the code. This is typical in stochastic
environments. If not desired, first a variable can be introduced that holds the generated random
number(s):
>> y = rand
y =
0.4186
>> y > 0.5
ans =
0
Other MATLAB functions work on vectors, but column-wise when called with matrix arguments.
An example is the function max, which returns the largest element of a vector:
>> rand(1)
ans =
0.8381
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>> rand(1,4)
ans =
0.9501
0.2311
0.6068
0.4860
>> rand(1,4)
ans =
0.8913
0.7621
0.4565
0.0185
0.1763
0.4057
>> max(rand(1,4))
ans =
0.8214
>> X = rand(1,6)
X =
0.9218
0.7382
0.9355
0.9169
>> max(x)
??? Undefined function or variable 'x'.
>> max(X)
ans =
0.9355
>> Y = rand(3,4)
Y =
0.4103
0.3529
0.8936
0.8132
0.0579
0.0099
0.1389
0.2028
0.1987
0.6038
0.2722
0.1988
>> max(Y)
ans =
0.8936
0.8132
0.2028
0.6038
>> max(Y')
ans =
0.6038
0.8936
0.1988
>> max(max(Y'))
ans =
0.8936
The most powerful functions in MATLAB are the matrix functions. Examples of such functions
are inv which returns the inverse of a matrix, and eig, which returns the eigenvalues and
eigenvectors of a matrix. To find out the right way a function should be used, you can use the help
command:
>> help eig
EIG
Eigenvalues and eigenvectors.
E = EIG(X) is a vector containing the eigenvalues of a square
matrix X.
[V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and
a full matrix V whose columns are the corresponding
eigenvectors so that X*V = V*D.
[V,D] = EIG(X,’nobalance’) performs the computation with
balancing disabled, which sometimes gives more accurate results
for certain problems with unusual scaling.
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MATLAB
E = EIG(A,B) is a vector containing the generalized eigenvalues
of square matrices A and B.
[V,D] = EIG(A,B) produces a diagonal matrix D of generalized
eigenvalues and a full matrix V whose columns are the
corresponding eigenvectors so that A*V = B*V*D.
See also CONDEIG.
Overloaded methods
help lti/eig.m
So, if we want to compute the eigenvalue decomposition , we can type the following:
>> [S Lambda] = eig(A);
To check that we made no errors, we can execute the following calculation:
>> S*Lambda*inv(S)
ans =
1.0000
2.0000
4.0000
5.0000
7.0000
8.0000
3.0000
6.0000
9.0000
Due to internal number rounding, the result equals to the matrix only up to a limited accuracy. In this
example we have
_
>> A - ans
ans =
1.0e-014 *
0.0777
0.2665
0.3553
-0.3109
0.0888
0.0888
-0.3997
0
0.3553
Take care when comparing computed results. A test such as
>> x == 0
can return a 0 (false), while
> abs(x) < 1e-15
can return 1 (true). Rational operators are discussed extensively in Part 2 of this tutorial.
MATLAB’s output format can be adjusted with the command format (see help format).
If we want to use, for example, a five digit floating-point representation, we type:
>> format short e
>> ans
ans =
7.7716e-16
2.6645e-15
3.5527e-15
-3.1086e-15
8.8818e-16
8.8818e-16
-3.9968e-15
0
3.5527e-15
It is also possible to suppress blank lines in the standard output.
To use a mathematical function without knowing its name in MATLAB, we can use the command
lookfor. This function searches all m-file help entries for a given string. How such m-files exactly
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look like will be discussed in detail in Part 3 of this tutorial. Assume we want to generate random
numbers. To find the corresponding MATLAB function, we can type:
>> lookfor 'random numbers'
RAND
Uniformly distributed random numbers.
RANDN Normally distributed random numbers.
RANDOM Generates random numbers from a named distribution.
and we can choose the one that specifically serves our purposes. Note that if the string contains more
than one word we must put quotes (‘.....’) around the string. For an overview of the most important
MATLAB functions commands and variables see [1, pp. 22-35].
1.4 Graphics
For the visualization of data we can open a graphical window. The command figure opens such a
window and returns a, for each window unique, integer (a so-called “handle”), for example
>> h = figure
h =
1
Using get(h)we can view the object properties of the figure (like position, dimension, etc.).
There are many ways to visualize data. The ones most used are plot for plotting two dimensional
data, and surf and mesh for three dimensional plots. See the available help information using the
help command. To plot, for example, a number of uniformly distributed outcomes, we can type:
>> n = 1:100;
>> X = rand(1,100);
>> plot(n,X);
Furthermore, we can adjust the plotting range and add a grid.
>> axis([0 200 -0.1 1.1]);
>> grid;
in octave use command
>> gset grid;
Several ways exist for plotting multiple functions at the same time with different colors and line
types. We can do this with only one plot statement
>> plot(n, X, ’r’, n, X>0.5, ’b’);
>> axis([0 200 -0.1 1.1]);
or separately using the command hold:
>>
>>
>>
>>
>>
>>
>>
Y = (X>0.5);
plot(X);
axis([0 100 -0.1 1.1]);
hold on
plot(Y, ’r’);
Z = Y.*X ;
plot(Z, ’g’);
In addition, you can generate two plots in different subplots in one figure (above or next to each
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MATLAB
other). To do this, use the command subplot. See the following example.
>>
>>
>>
>>
>>
>>
>>
>>
>>
subplot(3,1,1);
plot(X);
axis([0 100 -0.1 1.1]);
subplot(3,1,2);
plot(Y);
axis([0 100 -0.1 1.1]);
subplot(3,1,3);
plot(Z);
axis([0 100 -0.1 1.1]);
If desired, the colors of the subplots can be changed, so can the solid-line presentations. Such a
plotting could be:
>>
>>
>>
>>
>>
>>
>>
>>
>>
subplot(3,1,1);
plot(X, 'bo');
axis([0 100 -0.1 1.1]);
subplot(3,1,2);
plot(Y, 'r+');
axis([0 100 -0.1 1.1]);
subplot(3,1,3);
plot(Z, 'gd');
axis([0 100 -0.1 1.1]);
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