Matrices

Matlab is designed to make matrix manipulation as simple as possible. Every Matlab variable refers to a matrix [a number is a 1 by 1 matrix]. Start Matlab again, and enter the following command.

	>> a = [1,2,3; 4 5 6]

Note that:

the elements of a matrix being entered are enclosed by brackets;
a matrix is entered in "row-major order" [ie all of the first row, then all of the second row, etc];
rows are seperated by a semicolon [or a newline], and the elements of the row may be seperated by either a comma or a space.

The element in the i'th row and j'th column of a is referred to in the usual way:

	>> a(1,2), a(2,3)

It's very easy to modify matrices:

	>> a(2,3) = 10

The transpose of a matrix is the result of interchanging rows and columns. Matlab denotes the transpose by following the matrix with the single-quote [apostrophe].

	>> a'
	>> b=[1 1 1]'

New matrices may be formed out of old ones, in many ways. Enter the following commands; before pressing the enter key, try to predict their results!

	>> c = [a; 7 8 9]
	>> [a; a; a]
	>> [a, a, a]
	>> [a', b]
	>> [ [a; a; a], [b; b] ]

There are many built-in matrix constructions. Here are a few:

	>> rand(1,3), rand(2)
	>> zeros(3)
	>> ones(3,2)
	>> eye(3), eye(2,3)
	>> magic(3)
	>> hilb(5)

This last command creates the 5 by 5 "Hilbert matrix," a favorite example.

Use a semicolon to suppress output:

        >> s = zeros(20,25);

This is valuable, when working with large matrices. If you forget it, and start printing screenfuls of unwanted data, Control-C is Matlab's "break" key.

To get more information on these, look at the help pages for elementary and special matrices.

	>> help elmat
	>> help specmat

A central part of Matlab syntax is the "colon operator," which produces a list.

	>> -3:3

The default increment is by 1, but that can be changed.

	>> x = -3 : .3 : 3

This can be read: "x is the name of the list, which begins at -3, and whose entries increase by .3, until 3 is surpassed." You may think of x as a list, a vector, or a matrix, whichever you like.

You may wish use this construction to extract "subvectors," as follows.

	>> x(2:12)
	>> x(9:-2:1)

See if you can predict the result of the following.
[Hint: what will x(2) be? x(10)?].

	>> x=10:100;
	>> x(40:5:60)

The colon notation can also be combined with the earlier method of constructing matrices.

	>> a = [1:6 ; 2:7 ; 4:9]

A very common use of the colon notation is to extract rows, or columns, as a sort of "wild-card" operator which produces a default list. The following command produces the matrix a, followed by its first row [with all of its columns], and then its second column [with all of its rows].

	>> a, a(1,:), a(:,2)

        >> s = rand(20,5);  s(6:7, 2:9)

Matrices may also be constructed by programming. Here is an example, creating a "program loop."

	>> for i=1:10, 
	>> 	for j=1:10,
	>> 		t(i,j) = i/j;
	>> 	end
	>> end

There are actually two loops here, with one nested inside the other; they define t(1,1), t(1,2), t(1,3) ... t(1,10), t(2,1), t(2,2) ... , t(2,10), ... t(10,10) [in that order].

	>> t