COLON NOTATION

• One of the most important notations in Matlab is the "colon" notation. It has many different uses and we will likely not get a chance to discuss all of them.
• First we'll discuss the use of colons in building vectors. The general syntax is

v={begin}:{step}:{end}

Here are some examples,

x=0:.25:1   % x=[0, .25, .5, .75, 1]
x=1:-.25:0  % x=[1, .75, .5, .25, 0]
x=1:2:10    % x=[1, 3, 5, 7, 9]
B=A(:)      % stacks the columns of A.

So if A is 4 by 6, then B=A(:) is a column vector with 24 rows. You can reshape B with the command C=reshape(B,n,m) where the only restrictions are that n and m are positive integers with m*n=24. For example to rebuild A, type reshape(B,4,6).

A=[1 2 3 4;5 6 7 8; 9 10 11 12]
B=A(:)
transA=A'
C=transA(:)
D=reshape(C,6,2)
E=reshape(C,4,3)
F=reshape(B,3,4)

As another example, suppose that A is an n by m matrix and v is a column vector of length n*m. Then A(:)=v replaces the elements of A by the elements of v (by columns).

v=(-12:-1)'
A(:)=v

• The colon is also useful in describing submatrices of a matrix.
• the ith row is A(i,:)
• the jth column is A(:,j)
• the submatrix consisting of elements in the i through jth rows and p through qth columns is given by A(i:j,p:q).
• the submatrix consisting of elements in the i through jth rows is given by A(i:j,:).
• the submatrix consisting of elements in the p through qth columns is given by A(:,p:q).

A=[1 2 3 4;5 6 7 8; 9 10 11 12]
B=A(:,[2 4])
C=A([1 3],:)

• Another useful way to change a matrix is to use the empty matrix []. For example, the following syntax defines E to be the empty matrix

E=[]

The empty matrix can be used in many ways, for example
a) to initialize a matrix for later use and
b) to delete elements of a matrix A(i:j,:)=[] simply deletes the i through jth rows of A, i.e. this is the same as A=[A(1:(i-1),:);A((j+1):m,:)]

A(:,[2 4])=[]

As a final example we note that the statement B=A(:) in the first example in this lesson could have been done using a for loop (which we will talk about later) and the empty matrix as follows

y=[] 
for j=1:n, y=[y;A(:,j)],end   %we will get to loops later 

• As another example of colon notation and Hadamard products, lets consider elementary plotting commands.

x=0:.05:2*pi;
y1=sin(x); y2=cos(x);
plot(x,y1)  %this is the basic syntax for plotting a function 
            % f(x) at the values of a vector x. The graph is generated by linear
            %interpolation of the values of (x(j),f(x(j))) 
hold on     % this freezes the current axes for additional plotting
pause       % stop program execution until you hit any key
plot(x,y2)  % plot another function on the same axes
hold off    % release the axis so that future plots generate new axes

• Here is another way to obtain multiple plots

x=0:.05:2*pi;
y1=sin(x); y2=cos(x);
plot(x,y1,x,y2) 

You should look at the plot command in the help file since there are many, many commands that can be given to the plot statement. For example, to plot values y_j=f_j(x_j) for j=1, ...,n where each x_j is a vector, use the syntax plot(x_1,y_1,x_2,y_2,...,x_n,y_n). If the vectors x_j are the same column vector, say x=x_1= ... x_n, and the y_j=f_j(x), you can build the matrix Y=[y_1 y_2 ... y_n] and issue the command plot(x,Y).

x=(0:.05:1)';
Y=(x*ones(1,5)).^(ones(size(x))*(1:5)); % this gives a matrix whose columns are x^j, j=1..5
plot(x,Y)

• Sometimes you might want several graphs, each on its own figure window. This can be accomplished using the figure command

x=0:.05:2*pi;
y1=sin(x); y2=cos(x);
h1=figure
plot(x,y1,x,y2)

h2=figure
Y=[2*y1.*y2; sin(2*x)]; % Note the Hadamard operation in the first term 
plot(x,Y)        % Notice we  only have one graph since sin(2x) = 2 sin(x) cos(x)?

h3=figure
x=-2.5:.05:2.5;
y=x.^2.*(4-x.^2);
plot(x,y)

pause  % we insert a pause command to suspend program operation
close all  % the close command is used to close figure windows, 
           %close all simply closes all figure windows 

EXERCISES

1. Build the vector v=[100,95,90, ... ,-95,-100]
2. Build the vector v=[sin (pi),sin(2 pi), ... ,sin(10 pi)]
3. Build the 1 by 10 vector v=[0,1,1, ... ,1,1,0]
4. Build the vector v=[1^2,2^2, ... ,10^2]
5. Build the vector v=[2,4, ... ,2^20]
6. Build the vector with complex number entries v=[1+i,1+2i,1+3i, ... ,1+50i]. (hint: i= sqrt{-1} is built into matlab, remember 1+v adds a one to each entry of v)
7. Given the 6 by 9 matrix

A=[1 2 3 4 5 6 7 8 9;1 1 1 1 1 1 1 1 1;0 0 0 0 0 0 0 0 0; ...
9 8 7 6 5 4 3 2 1;-1 -1 -1 -1 -1 -1 -1 -1 -1;2 1 2 1 2 1 2 1 1]

1. Build the submatrix B consisting of the first and fourth rows and the third thru sixth columns of A.
2. Build the matrix C consisting of elements of the even numbered rows and the odd numbered columns of A.
3. Reshape the matrix A to a new matrix D which is 2 by 27.
4. Build a new matrix E consisting of the second and the negative of the fifth rows of A.
5. Use the empty matrix to omit the first, second and fourth thru sixth rows of A.
6. Build the matrix F consisting of the columns of A in reverse order. hint: given the vector

v=[1 2 3 4 5]
u=v(5:-1:1)

You might also look at help fliplr.

7. By simply reassigning and reversing the fourth row of A make the first and fourth row of A the same. hint: given the matrix

M=[1 0 1 0 1;1 1 1 1 1;-2 -2 -2 -2 -2;1 2 3 4 5]
M(4,:)=-M(2,:)

8. Create a matrix G which consists of the matrix A with all the entries in the even numbered rows replaced by zeros.
9. Build a 8 by 11 matrix H which is the matrix A bordered on all four sides with ones.
8. Build the 1 by 200 vector v=[0,1,0,2,0, ... ,0,99,0,100] (hint: you can assign the values of a matrix by

v([1 3 5])=1:3

and the values v(2) and v(4) are automatically set equal to 0.

9. Given an interval (a,b) it is often useful to build a partition consisting of n disjoint intervals [x(j),x(j+1)] of length 1/n. This can be accomplished using n+1 equidistributed points in the interval. A set of such points is obtained from x(j)=a+(b-a)*((j-1)/n) where j runs from 1 to (n+1). Given the interval [-7,9] use Matlab to build this vector x when n=30.
10. On the same axes plot sin(j*pi*x) for j=1..5. Use x=0:.1:2*pi.
11. Plot sin(x)/x on -5*pi to 5*pi
12. Plot x sin(1/x) for x in the interval [0.01 ,5].
    hint: use

x1=.01:.0025:.25; x2=[x1 .3:.025:.5];