More Tutorials:

Front Page

vectors

matrices

vector operations

loops

plots

executable files

subroutines

if statements

data files

Front Page

vectors

matrices

vector operations

loops

plots

executable files

subroutines

if statements

data files

A basic introduction to defining and manipulating matrices is given here. It is assumed that you know the basics on how to define and manipulate vectors using matlab.

Defining a matrix is similar to defining a vector. To define a matrix, you can treat it like a column of row vectors (note that the spaces are required!):

>> A = [ 1 2 3; 3 4 5; 6 7 8] A = 1 2 3 3 4 5 6 7 8

You can also treat it like a row of column vectors:

>> B = [ [1 2 3]' [2 4 7]' [3 5 8]'] B = 1 2 3 2 4 5 3 7 8

(Again, it is important to include the spaces.)

If you have been putting in variables through this and the tutorial on vectors,
then you probably have a lot of variables defined. If you lose track of
what variables you have defined, the *whos* command will let you
know all of the variables you have in your work space.

>> whos Name Size Bytes Class A 3x3 72 double array B 3x3 72 double array v 1x5 40 double array Grand total is 23 elements using 184 bytes

We assume that you are doing this tutorial after completing the previous tutorial. The vector v was defined in the previous tutorial.

As mentioned before, the notation used by Matlab is the standard linear algebra notation you should have seen before. Matrix-vector multiplication can be easily done. You have to be careful, though, your matrices and vectors have to have the right size!

>> v = [0:2:8] v = 0 2 4 6 8 >> A*v(1:3) ??? Error using ==> * Inner matrix dimensions must agree. >> A*v(1:3)' ans = 16 28 46

Get used to seeing that particular error message! Once you start throwing matrices and vectors around, it is easy to forget the sizes of the things you have created.

You can work with different parts of a matrix, just as you can with vectors. Again, you have to be careful to make sure that the operation is legal.

>> A(1:2,3:4) ??? Index exceeds matrix dimensions. >> A(1:2,2:3) ans = 2 3 4 5 >> A(1:2,2:3)' ans = 2 4 3 5

Once you are able to create and manipulate a matrix, you can perform many standard operations on it. For example, you can find the inverse of a matrix. You must be careful, however, since the operations are numerical manipulations done on digital computers. In the example, the matrix A is not a full matrix, but matlab's inverse routine will still return a matrix.

>> inv(A) Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 4.565062e-18 ans = 1.0e+15 * -2.7022 4.5036 -1.8014 5.4043 -9.0072 3.6029 -2.7022 4.5036 -1.8014

By the way, Matlab is case sensitive. This is another potential source of problems when you start building complicated algorithms.

>> inv(a) ??? Undefined function or variable a.

Other operations include finding an approximation to the eigen
values of a matrix. There are two versions of this routine, one
just finds the eigen values, the other finds both the eigen values
and the eigen vectors. If you forget which one is which, you can
get more information by typing *help eig* at the matlab
prompt.

>> eig(A) ans = 14.0664 -1.0664 0.0000 >> [v,e] = eig(A) v = -0.2656 0.7444 -0.4082 -0.4912 0.1907 0.8165 -0.8295 -0.6399 -0.4082 e = 14.0664 0 0 0 -1.0664 0 0 0 0.0000 >> diag(e) ans = 14.0664 -1.0664 0.0000

There are also routines that let you find solutions to equations. For example, if Ax=b and you want to find x, a slow way to find x is to simply invert A and perform a left multiply on both sides (more on that later). It turns out that there are more efficient and more stable methods to do this (L/U decomposition with pivoting, for example). Matlab has special commands that will do this for you.

Before finding the approximations to linear systems, it is important to remember that if A and B are both matrices, then AB is not necessarily equal to BA. To distinguish the difference between solving systems that have a right or left multiply, Matlab uses two different operators, "/" and "\". Examples of their use are given below. It is left as an exercise for you to figure out which one is doing what.

>> v = [1 3 5]' v = 1 3 5 >> x = A\v Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 4.565062e-18 x = 1.0e+15 * 1.8014 -3.6029 1.8014 >> x = B\v x = 2 1 -1 >> B*x ans = 1 3 5 >> x1 = v'/B x1 = 4.0000 -3.0000 1.0000 >> x1*B ans = 1.0000 3.0000 5.0000

Finally, sometimes you would like to clear all of your data
and start over. You do this with the "clear" command.
Be careful though, it does not ask
you for a second opinion and its results are **final**.

>> clear >> whos