Thursday, 24 April 2014

MATLAB TUTORIAL : VECTORS AND MATRICES

VECTORS
MATLAB is based on matrix and vector algebra. So, even scalars are treated as 1×1 matrices.
We have two ways to define vectors:
  1. Arbitrary element (not equally spaced elements):
    >> v = [1 2 5];
    
    This creates a 1×3 vector with elements 1, 2 and 5. Note that commas could have been used in place of spaces to separate the elements, like this:
     
    >> v = [1, 2, 5];
    
    Keep in mind that we the index of the first element is 1 not 0:
    >> v(0)
    Subscript indices must either be real positive integers or logicals.
    >> v(1)
    ans =
         1
    
    If we want to add more elements:
    >> v(4) = 10;
    
    The yields the vector v=[1,2,5,10].
    We can also use previously defined vector:
    >> >> w = [11 12];
    >> x = [v,w]
    
    Now, x=[1,2,5,10,11,12,]
  2. Equally spaced elements :
    >> t = 0 : .5 : 3
    
    This yields t=[0 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000] which is a 1×7 vector. If we give only two numbers, then the increment is set to a default of 1:
    >> t = 0:3;
    
    This will creates a 1×4 vector with the elements 0, 1, 2, 3.
One more, adding vectors:
>> u = [1 2 3];
>> v = 4 5 6];
>> w = u + v
w =
     5     7     9


COLUMN VECTORS
We can create column vectors. In Matlab, we use semicolon(';') to separate columns:
>> RowVector = [1 2 3]   % 1 x 3 matrix
RowVector =
     1     2     3

>> ColVector = [1;2;3]   % 3 x 1 matrix
ColVector =
     1
     2
     3

>> M = [1 2 3; 4 5 6]    % 2 x 3 matrix

M =

     1     2     3
     4     5     6


MATRICES
Matrices are defined by entering the elements row by row:
>> M = [1 2 3; 4 5 6]
M =
     1     2     3
     4     5     6

There are a number of special matrices:

null matrixM = [ ];
mxn matrix of zerosM = zeros(m,n);
mxn matrix of onesM = ones(m,n);
nxn identity matrixM = eyes(n);

If we want to assign a new value for a particular element of a matrix (2nd row, 3rd column):
M(2,3) = 99;


ACCESSING MATRIX ELEMENT
We can access an element of a matrix in two ways:
  • M(row, column)
  • M(n-th_element)
>> M = [10 20 30 40 50; 60 70 80 90 100];
>> M
M =
    10    20    30    40    50
    60    70    80    90   100
 
>> M(2,4) % 2nd row 4th column
ans =
    90

>> M(8)  % 8th element 10, 60, 20, 70, 30, 80, 40, 90...
ans =
    90
Also, we can access several elements at once:
>> M(1:2, 3:4)  % row 1 2, column 3 4
ans =
    30    40
    80    90
>> M(5:8)   % from 5th to 8th elements
ans =
    30    80    40    90


ELEMENT BASED OPERATION USING DOT('.')
If we use dot('.') on an operation, it means do it for all elements. The example does ^3 for all the elements of a matrix:
>> M = [1 2 3; 4 5 6; 7 8 9];
>> M
M =
     1     2     3
     4     5     6
     7     8     9

>> M.^3
ans =
     1     8    27
    64   125   216
   343   512   729
We can get inverse of each element:
>> 1./M
ans =
    1.0000    0.5000    0.3333
    0.2500    0.2000    0.1667
    0.1429    0.1250    0.1111


INVERSE MATRIX
We get inverse of a matrix using inv():
>> M = [1 2; 3 4]
M =
     1     2
     3     4

>> inv(M)
ans =
   -2.0000    1.0000
    1.5000   -0.5000


TRANSPOSE MATRIX USING A SINGLE QUOTE(')
We get transpose of a matrix using a single quote("'"):
>> M = [1 2 3; 4 5 6; 7 8 9]
M =
     1     2     3
     4     5     6
     7     8     9

>> M'
ans =
     1     4     7
     2     5     8
     3     6     9




FLIPUD() AND FLIPLR()
flipud(): flip up/down:
>> M = [1 2 3; 4 5 6; 7 8 9]
M =
     1     2     3
     4     5     6
     7     8     9

>> flipud(M)
ans =
     7     8     9
     4     5     6
     1     2     3

fliplr(): flip left/right:
>> M
M =
     1     2     3
     4     5     6
     7     8     9

>> fliplr(M)
ans =

     3     2     1
     6     5     4
     9     8     7




ELEMENT ROTATION
We can do rotate elements, for instance, 90 degree counter-clock wise:
>> M

M =

     1     2     3
     4     5     6
     7     8     9

>> rot90(M)

ans =

     3     6     9
     2     5     8
     1     4     7




RESHAPE()
We can make a change to the number of rows and columns as far as we keep the total number of elements:
>> M = [1 2 3 4; 5 6 7 8; 9 10 11 12]
M =
     1     2     3     4
     5     6     7     8
     9    10    11    12

>> reshape(M, 2, 6)  % Convert M to 2x6 matrix
ans =
     1     9     6     3    11     8
     5     2    10     7     4    12




VECTOR & FUNCTION
Functions are applied element by element:
>> t = 0:100;
>> f = cos(2*pi*t/length(t))
>> plot(t,f)
f=cos(2πt/length(t)) creates a vector f with elements equal to 2πt/length(t) for t=0,1,2,...,100.

Cosine.png 

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