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MatLab Notes

MatLab notes

Enter a magic square from the Renaissance engraving Melancholia I by the
German artist and amateur mathematician Albrecht Dürer.

A = [16 3 2 13; 5 10 11 8; 9 6 7 12; 4 15 14 1]

sum(A)   returns sums of columns (all 4 = 34)

A'  is transpose of A; result is implicity put in variable ans

sum(A')'  produces column of 34s - the result of transposing column sums on A'

diag(A)  produces diagonal terms list

sum(diag(A))  sums diag terms = 34

It is also possible to refer to the elements of a matrix with a single subscript, A(k). This is the usual way of
referencing row and column vectors. But it can also apply to a fully two-dimensional matrix, in which case the
array is regarded as one long column vector formed from the columns of the original matrix. So, for our magic
square, A(8) is another way of referring to the value 15 stored in A(4,2).

A = magic(3)

eig(A)
poly(A)
inv(A)
ans * A

life

simple algebra (neeb,noob,nub problem of 6-3-2003)
A = [1 1 0; 1 0 1; 0 1 1]
b = inv(A) -> 
b =
    0.5000    0.5000   -0.5000
    0.5000   -0.5000    0.5000
   -0.5000    0.5000    0.5000

r = [3 4 5]
b * r' ->
ans =
     1
     2
     3
A * b ->
ans =
     1     0     0
     0     1     0
     0     0     1



From math.utah.edu, Matlab tutorial
Finally, we will do a little piece of programming. Let a be the matrix 

              0.8  0.1
              0.2  0.9

and let x be the column vector 
              1
              0

We regard x as representing (for example) the population state of an island. The 
first entry (1) gives the fraction of the population in the west half of the 
island, the second entry (0) give the fraction in the east half. The state of 
the population T units of time later is given by the rule y = ax. This expresses 
the fact that an individual in the west half stays put with probability 0.8 and 
moves east with probability 0.2 (note 0.8 + 0.2 = 1), and the fact that in 
individual in the east stays put with probability 0.9 and moves west with 
probability 0.1. Thus, successive population states can be predicted/computed by 
repeated matrix multiplication. This can be done by the following Matlab 
program: 

       >> a = [ 0.8 0.1; 0.2 0.9 ]
       >> x = [ 1; 0 ]
       >> for i = 1:20, x = a*x, end

What do you notice? Is there an explanation? Is there a lesson to be learned? 



To make a graph of y = sin(t) on the interval t = 0 to t = 10 we do the following: 

  >> t = 0:.3:10;
  >> y = sin(t);
  >> plot(t,y)

The command t = 0:.3:10; defines a vector with components ranging from 0 to 10 
in steps of 0.3. The y = sin(t); defines a vector whose components are sin(0), 
sin(0.3), sin(0.6), etc. Finally, plot(t,y) use the vector of t and y values to 
construct the graph. 



Functions of two variables 

Here is how we graph the fuction z(x,y) = x exp( - x^2 - y^2): 

   >> [x,y] = meshdom(-2:.2:2, -2:.2:2);
   >>  z = x .* exp(-x.^2 - y.^2);
   >>  mesh(z)

The first command creates a matrix whose entries are the points of a grid in the 
square -2 <= x <= 2, -2 <= y <= 2. The small squares which make up the grid are 
0.2 units wide and 0.2 unit tall. The second command creates a matrix whose 
entries are the values of the function z(x,y) at the grid points. The third 
command uses this information to construct the graph.