introduction to MATLAB

This is an interactive introduction to MATLAB. I have provided a sequence of commands for you to type in. The designation RET means that you should type the "return" key; this is implicit after a command.

To bring up MATLAB from from the operating system prompt 

lab%

you should type matlab

lab% matlab RET

This will present the prompt

>>

You are now in MATLAB.
If you are using the X Window system on the Mathematics Department workstations then you can also start MATLAB from the Main Menu by selecting "matlab" from the "Math Applications" submenu. A window should pop up and start MATLAB. When you run MATLAB under the window system, whether you start from the menu or a system prompt, a small MATLAB logo window will pop up while the program is loading and disappear when MATLAB is ready to use.
When you are ready to leave, type exit

>> exit RET 

In the course of the tutorial if you get stuck on what a command means type

>> help <command name> RET

and then try the command again.
You should record the outcome of the commands and experiments in a notebook.
Remark: Depending on the Info reader you are using to navigate this tutorial, you might be able to cut and paste many of the examples directly into Matlab.

Building Matrices

Matlab has many types of matrices which are built into the system. A 7 by 7 matrix with random entries is produced by typing

rand(7) 

You can generate random matrices of other sizes and get help on the rand command within matlab:

rand(2,5)

help rand

Another special matrix, called a Hilbert matrix, is a standard example in numerical linear algebra.

hilb(5)

help hilb

A 5 by 5 magic square is given by the next command:

magic(5)

help magic

A magic square is a square matrix which has equal sums along all its rows and columns. We'll use matrix multiplication to check this property a bit later.
Some of the standard matrices from linear algebra are easily produced:

eye(6)

zeros(4,7)

ones(5)

You can also build matrices of your own with any entries that you may want.

[1  2   3   5   7   9]

[1, 2, 3; 4, 5, 6; 7, 8, 9]

[1  2 RET  3  4 RET 5  6]

[Note that if you are using cut-and-paste features of a window system or editor to copy these examples into Matlab then you should not use cut-and-paste and the last line above. Type it in by hand, touching the Return or Enter key where you see RET, and check to see whether the carriage returns make any difference in Matlab's output.]
Matlab syntax is convenient for blocked matrices:

[eye(2);zeros(2)]

[eye(2);zeros(3)]

[eye(2),ones(2,3)]

Did any of the last three examples produce error messages? What is the problem?

Variables

Matlab has built-in variables like pi, eps, and ans. You can learn their values from the Matlab interpreter.

pi

eps

help eps

At any time you want to know the active variables you can use who:

who

help  who

The variable ans will keep track of the last output which was not assigned to another variable.

magic(6)

ans

x = ans

x = [x, eye(6)]

x

Since you have created a new variable, x, it should appear as an active variable.

who

To remove a variable, try this:

clear x

x

who

Functions

a = magic(4)

Take the transpose of a:

a'

Note that if the matrix A has complex numbers as entries then the Matlab function taking A to A' will compute the transpose of the conjugate of A rather than the transpose of A.
Other arithmetic operations are easy to perform.

3*a

-a

a+(-a)

b = max(a)

max(b)

Some Matlab functions can return more than one value. In the case of max the interpreter returns the maximum value and also the column index where the maximum value occurs.

[m, i] = max(b)

min(a)

b = 2*ones(a)

a*b

a

We can use matrix multiplication to check the "magic" property of magic squares.

A = magic(5)

    b = ones(5,1)

    A*b

    v = ones(1,5)

    v*A

Matlab has a convention in which a dot in front of an operation usually changes the operation. In the case of multiplication, a.*b will perform entry-by-entry multiplication instead of the usual matrix multiplication.

a.*b  (there is a dot there!)

x = 5

x^2

a*a

a^2

a.^2   (another dot)

a

triu(a)

tril(a)

diag(a)

diag(diag(a))

c=rand(4,5)

size(c)

[m,n] = size(c)

m

d=.5-c

There are many functions which we apply to scalars which Matlab can apply to both scalars and matrices.

sin(d)

exp(d)

log(d)

abs(d)

Matlab has functions to round floating point numbers to integers. These are round, fix, ceil, and floor. The next few examples work through this set of commands and a couple more arithmetic operations.

f=[-.5 .1 .5]

round(f)

fix(f)

ceil(f)

floor(f)

sum(f)

prod(f)

Relations and Logical Operations

In this section you should think of 1 as "true" and 0 as "false." The notations &, |, ~ stand for "and,""or," and "not," respectively. The notation == is a check for equality.

a=[1 0 1 0]

b=[1  1  0  0]

a==b

a<=b

~a

a&b

a & ~a

a | b

a | ~a

There is a function to determine if a matrix has at least one nonzero entry, any, as well as a function to determine if all the entries are nonzero, all.

a

any(a)

c=zeros(1,4)

d=ones(1,4)

any(c)

all(a)

all(d)

e=[a',b',c',d']

any(e)

all(e)

any(all(e))

Colon Notation

Matlab offers some powerful methods for creating arrays and for taking them apart.

x=-2:1

length(x)

-2:.5:1

-2:.2:1

a=magic(5)

a(2,3)

Now we will use the colon notation to select a column of a.

a(2,:)

a(:,3)

a

a(2:4,:)

a(:,3:5)

a(2:4,3:5)

a(1:2:5,:)

You can put a vector into a row or column position within a.

a(:,[1 2 5])

a([2 5],[2 4 5])

You can also make assignment statements using a vector or a matrix.

b=rand(5)

b([1 2],:)=a([1 2],:)

a(:,[1 2])=b(:,[3 5])

a(:,[1 5])=a(:,[5 1])

a=a(:,5:-1:1)

When you a insert a 0-1 vector into the column position then the columns which correspond to 1's are displayed.

v=[0 1 0 1 1]

a(:,v)

a(v,:)

This has been a sample of the basic MATLAB functions and the matrix manipulation techniques. At the end of the tutorial there is a listing of functions. The functions that you have available will vary slightly from version to version of MATLAB. By typing

help

you will get access to descriptions of all the Matlab functions.

Miscellaneous Features

You may have discovered by now that MATLAB is case sensitive, that is

"a" is not the same as "A." 

If this proves to be an annoyance, the command

casesen

will toggle the case sensitivity off and on.
The MATLAB display only shows 5 digits in the default mode. The fact is that MATLAB always keeps and computes in a double precision 16 decimal places and rounds the display to 4 digits. The command

format long

will switch to display all 16 digits and

format short

will return to the shorter display. It is also possible to toggle back and forth in the scientific notation display with the commands

format  short e

and

format long e

It is not always necessary for MATLAB to display the results of a command to the screen. If you do not want the matrix A displayed, put a semicolon after it, A;. When MATLAB is ready to proceed, the prompt >> will appear. Try this on a matrix right now.
Sometimes you will have spent much time creating matrices in the course of your MATLAB session and you would like to use these same matrices in your next session. You can save these values in a file by typing

save filename

This creates a file

filename.mat 

which contains the values of the variables from your session. If you do not want to save all variables there are two options. One is to clear the variables off with the command

clear a b c

which will remove the variables a,b,c. The other option is to use the command

save x y z

which will save the variables x,y,z in the file filename.mat. The variables can be reloaded in a future session by typing

load filename

When you are ready to print out the results of a session, you can store the results in a file and print the file from the operating system using the "print" command appropriate for your operating system. The file is created using the command

diary filename

Once a file name has been established you can toggle the diary with the commands

diary on

and

diary off

This will copy anything which goes to the screen (other than graphics) to the specified file. Since this is an ordinary ASCII file, you can edit it later. Discussion of print out for graphics is deferred to the project "Graphics" where MATLAB's graphics commands are presented.
Some of you may be fortunate enough to be using a Macintosh or a Sun computer with a window system that allows you to quickly move in and out of MATLAB for editing, printing, or other processes at the system level. For those of you who are not so fortunate, MATLAB has a feature which allows you to do some of these tasks directly from MATLAB. Let us suppose that you would like to edit a file named myfile.m and that your editor executes on the command ed. The MATLAB command

!ed myfile.m

will bring up your editor and you can now work in it as you usually would. Obviously the exclamation point is the critical feature here. When you are done editing, exit your editor as you usually would, and you will find that you are back in your MATLAB session. You can use the ! with many operating system commands.